Rocket In My Pocket

If you're Pe is below the Mun's surface on approach before you do your retrograde capture burn, you need to burn radial (light blue circle) most likely out, not prograde.
I fired up the tutorial and that's what it says as well, perhaps you misread it?

I use and recommend the in game Transfer Window Planner mod. It works fine in the latest version so no worries there.
Here's a (very) crash course in using it I wrote up for someone on Steam a while ago.
Ok, so crash course in TWP time. I learned just by poking at it for a bit, most of the information is conveyed visually so it's pretty simple, despite all the text/numbers.

1. I've got us an example transfer to Duna from Kerbin setup. 100k to 100k. The most important info to take from this is that we'll be departing on year 12, day 71.


2. While in map view we get buttons to show us both our Phase Angles and our Ejection Angle.
The Phase angle isn't super important for you to worry about, it'll be aligned all by itself on the proper departure day.


The ejection angle is important though. (Not right now so much, but it will be.)


3. Now, I've fast forwarded to the day of departure. (Well one day off but that won't matter lol.) As you can see, our Phase angles are aligned (The green and blue lines pointing at Duna), which means this is the cheapest time to go to Duna.


4.Now the Ejection angle, the blue line with the green arrow coming off it tells us where to put our maneuver node and which direction to burn, (Typically prograde.)

5. Now by simply adding some prograde thrust to our maneuver node we get a duna encounter easily, because we left at the right time, going the right way.

Hope that helped, (and I can copy-pasta it elsewhere to teach others.) Let me know if you have any more questions!

Welcome to the forums!
The launcher was "discontinued" awhile ago afaik, it's fairly non-functional now; just ignore it.
The permission request is due to your security settings. You could make a shortcut to KSP.exe and set it to administrator authority, should stop the prompt I think? Anyways, nothing to be concerned about, just your computer being over zealous.

You should get Transfer Window Planner mod too, and check a DV map. 16,000Dv?! That's supreme overkill for Duna.
For comparison here's an early Career Duna flyby and return mission. Only has like 6k Dv. (Whoops, looking at it in the editor closer just now, and realized I was sending the top bit up unfueled, so I must have fueled it in orbit. So...roughly 9k Dv then. Still, a lot cheaper! Also a good way to save on launches as a side note/tip, send the mission critical bit up unfueled then rendezvous and fuel it up before it leaves.)
Seriously though, congrats on getting a fly by in the first place! Very impressive without any external help, besides KER. 
Transfer Window Planner:
Delta-V map:
 

In the Contracts folder in Gamedata there is a Contracts.cfg file.
I believe this is the section you are looking for, I have bolded/italicized the most relevant lines.
Best of luck!
Satellite // Satellite Deployment Contracts
    {
        MaximumAvailable = 9999 // The maximum amount of inactive satellite contracts in Mission Control
        MaximumActive = 9999 // The maximum amount of active satellite contracts that have been accepted
        AnimationDuration = 8 // How long it takes the orbit render animation to loop
        ContextualChance = 50 // The maximum chance for a satellite adjustment or network request
        ContextualAssets = 2 // The amount of satellites required on non-home planets to reach that chance
        ContextualHomeAssets = 4 // The amount of satellites required on the home planet to reach that chance
        NetworkMinimum = 3 // The minimum amount of satellites that are considered a network
        NetworkMaximum = 4 // The maximum amount of satellites that are considered a network
        MinimumDeviationWindow = 750 // The lowest that an altitude deviation can get at the periapsis
        TrivialDeviation = 7 // The percent difference allowed for altitude and relevant angles to trigger easy contracts
        SignificantDeviation = 5 // The percent difference allowed for altitude and relevant angles to trigger medium contracts
        ExceptionalDeviation = 3 // The percent difference allowed for altitude and relevant angles to trigger hard contracts
        TrivialAltitudeDifficulty = 0.2 // A ratio representing how eccentric orbits can get on easy contracts
        TrivialInclinationDifficulty = 0.2 // A ratio representing how inclined orbits can get on easy contracts
        SignificantAltitudeDifficulty = 0.4 // A ratio representing how eccentric orbits can get on medium contracts
        SignificantInclinationDifficulty = 0.4 // A ratio representing how inclined orbits can get on medium contracts
        ExceptionalAltitudeDifficulty = 0.8 // A ratio representing how eccentric orbits can get on hard contracts
        ExceptionalInclinationDifficulty = 0.8 // A ratio representing how inclined orbits can get on hard contracts
        PreferHome = True // Whether to utilize home override chances to prefer home satellites
        AllowSolar = True // Whether to allow satellite requests for solar orbits at all
        AllowEquatorial = True // Whether to allow equatorial orbits at all
        AllowPolar = True // Whether to allow polar orbits at all
        AllowSynchronous = True // Whether to allow synchronous orbits at all
        AllowStationary = True // Whether to allow stationary orbits at all
        AllowTundra = True // Whether to allow tundra orbits at all
        AllowKolniya = True // Whether to allow Molniya orbits at all
        Expiration
        {
            MinimumExpireDays = 1 // The minimum time in days the contract will remain in Mission Control before expiring
            MaximumExpireDays = 7 // The maximum time in days the contract will remain in Mission Control before expiring
            DeadlineDays = 2130 // How many days the player gets to complete the contract
        }
        Funds
        {
            BaseAdvance = 16000 // The base funds advance before planet and difficulty modifiers
            BaseReward = 41000 // The base funds completion reward before planet and difficulty modifiers
            BaseFailure = 0 // The base funds failure penalty before planet and difficulty modifiers
            HomeMultiplier = 2 // How much funds rewards are multiplied by on satellite requests at home
            PolarMultiplier = 1 // How much funds rewards are multiplied by on polar satellite requests
            SynchronousMultiplier = 1.1 // How much funds rewards are multiplied by on synchronous satellite requests
            StationaryMultiplier = 1.2 // How much funds rewards are multiplied by on stationary satellite requests
            TundraMultiplier = 1.2 // How much funds rewards are multiplied by on tundra satellite requests
            KolniyaMultiplier = 1.2 // How much funds rewards are multiplied by on Molniya satellite requests
        }
        Science
        {
            BaseReward = 0 // The base science completion reward before difficulty modifiers
            HomeMultiplier = 0.5 // How much science rewards are multiplied by on satellite requests at home
            PolarMultiplier = 1 // How much science rewards are multiplied by on polar satellite requests
            SynchronousMultiplier = 1.1 // How much science rewards are multiplied by on synchronous satellite requests
            StationaryMultiplier = 1.2 // How much science rewards are multiplied by on stationary satellite requests
            TundraMultiplier = 1.2 // How much science rewards are multiplied by on tundra satellite requests
            KolniyaMultiplier = 1.2 // How much science rewards are multiplied by on Molniya satellite requests
        }
        Reputation
        {
            BaseReward = 10 // The base reputation completion reward before difficulty modifiers
            BaseFailure = 5 // The base reputation failure penalty before difficulty modifiers
            HomeMultiplier = 1.25 // How much reputation rewards are multiplied by on satellite requests at home
            PolarMultiplier = 1 // How much reputation rewards are multiplied by on polar satellite requests
            SynchronousMultiplier = 1.1 // How much reputation rewards are multiplied by on synchronous satellite requests
            StationaryMultiplier = 1.2 // How much reputation rewards are multiplied by on stationary satellite requests
            TundraMultiplier = 1.2 // How much reputation rewards are multiplied by on tundra satellite requests
            KolniyaMultiplier = 1.2 // How much reputation rewards are multiplied by on Molniya satellite requests

Gonna keep this more of a simple outline rather than a comprehensive guide as I don't like to spoil the reward of learning the finer points yourself. Feel free to ask follow up questions though!
Step 1: Be in orbit.
Step 2: Think about where the Mun will be, not where it is. It continues to travel even while you are on the way so you'll need to aim ahead of it. (Roughly 1/5 to 1/4 of it's orbit.)
Step 3: Setup your maneuver node on the opposite side of your orbit from the Mun. Ie. The side furthest from the Mun. This is because burning pro-grade here will cause the other side to grow outwards. (Setting the Mun as your target can help but isn't required. This is more important when inclination comes into play, but the Mun and Kerbin have the same inclination.)
Step 4: Pull the pro-grade handle until your orbit overlaps the Mun's. Now play with the radial or move the actual node itself back and forth a bit until you get a Mun intercept. (You'll see a new purple line and a purple Mun indicator appear.)
Step 5: Wait till your node and execute it. When done, cut engines and drift to the Mun encounter; time warp if desired.
Step 5: Assuming you don't want to just do a fly by; burn retrograde at Mun Pe until your Mun Ap matches and you are circular.
Step 6: Congrats, you're orbiting the Mun!

If you haven't done a fresh install since it was changed to Tilde, it'll still be Backspace for you.
Afaik updating the game doesn't change your settings.cfg which is why.

Welcome to the forums @dairwin!
As @Geonovast said, if you have it on Steam it's as simple as right clicking KSP in your Steam library, selecting properties,  going to the local files tab, then hitting "browse local files."
You can either upload the actual .craft file to a free file sharing site like Dropbox, or move it on USB stick or something like that.

Welcome to the forums!
Here's a link to the the exact computer I used to play KSP on, and it was more than adequate. It's 550$ currently but I got it on sale awhile back for a bit less than that. Meet's or exceeds all your requirements.
http://www.dell.com/en-us/shop/dell-desktop-computers/inspiron-desktop/spd/inspiron-3656-desktop/ddccz301b

I've been seeing this problem quite a bit lately.
Lot's of people have had success with setting their system resolution lower than their native resolution.
Ie. Set your computers resolution to something smaller than the actual size of your monitor.

Is the game making an actual crash log? It'll be in the KSP folder, in a folder named as a date.
Is this the first time you've tried to install and play KSP on this computer or have you launched it successfully before? You've never used any mods? Or you just deleted them?
Are your graphics card drivers updated?

Ok, so I sorted it out myself...feel a little silly for making this post now lol.
I'll detail the cause and solution though in case this happens to anyone else.
Apparently it was a scaling issue; I'm playing KSP at 4k now, and at 150% scale the icons on the navball start disappearing when you are over them, decreasing the Navball scale to 140% fixed the problem. In a similar fashion the Altitude meter no longer displays numbers at 150% scale but reducing it to 140% fixed that as well. I have no clue why this is personally. However my problem is solved for now and I'm back to playing happily so; all's well that ends well. Lol.

Well, looking at the specs on Google; it's not terrible besides the 4 Gb of RAM.
You say it plays KSP, so yeah; it's a good computer to run KSP in my opinion.
I used to play KSP on an old laptop that wasn't much better, and I enjoyed it just as much then on the lowest settings at 20fps as I do now on a better computer with highest settings and 60fps.

Yeah, I don't think it's your hardware showing cracks so much as it is the game itself.
Also; I'm pretty sure the game still has the long running memory leak on craft revert it's always had as well. Which will tend to increase RAM load as you play over long sessions. (Especially if you do a lot of reverting while testing like I do...)
For now, just restart the game every hour or two. That's the strategy I've adopted for now anyways lol. 

Here's another mod that may prove helpful to you.
 

You'd need to be going less than 250 m/s to safely deploy the chutes. (500 m/s for Drogues.)
The reason they all stayed red was to let you know it wasn't safe to open them yet, basically you were still going way to fast.
http://wiki.kerbalspaceprogram.com/wiki/Parachute

-The placement of the plane when spawned is based on it's placement relative to the floor of the SPH.
-You'll have to more specific than "wonky" or I can't help you. Also, what do you mean by "flaps"? Flaps are for landing/takeoff only. Do you mean aerodynamic control surfaces in general?
-You are using radial symmetry mode then, switch to mirror symmetry mode and they will work fine.
-On PC holding the "Alt" key enables node snap mode which will fix your problem, no clue what the key on Xbox is.
So yeah, none of these are really bugs.

Sounds like you are using radial symmetry. If you use radial for wheels, you'll encounter this problem a lot.
Press "R" to switch to mirror symmetry, this will make the wheels work correctly.
(Basically if they are "radial" they will be placed "opposite" from each other which is bad for things like which direction the wheel spins.)

Not to burst your bubble but there's a hard coded limit to how big something can get given that the physics engine cuts out about 2.5km from the currently controlled craft.
Of course you could  make it up of separate vessels sort of "floating" near each other but I'm not sure how well this will work out. The sheer number of parts involved is likely to turn even the most impressive computer into a slideshow.
I like your ambition, but I think you've bitten off more than you or anyone else can chew. I don't think an actual Dyson sphere is possible in KSP (or in real life for that matter.) Maybe go for a Dyson "swarm" or "ring" instead?

Myself and others have been having an issue with the Klaw since 1.1.3
Doesn't seem like this bug is affecting everyone, as I haven't heard much complaining about it.

Yes.
Yes it is.
This should really go in "Technical Support" Modded or Unmodded as the case may be. Someone there may be able to help you. Best of luck!

Tried loading it up, same things happens every time. Giant explosion.
Even rebuilding the fairing several different ways didn't help. Some part of the ship always collides with it and explodes, despite never actually touching it or moving in any way.
In the flight log I'm seeing some references to strut connectors colliding but I've looked the whole ship over and I don't see that you actually used any. I'm getting the feeling this has something to do with the invisible "auto-strutting" the fairings use. Only advice I can give you is to leave the fairing behind for now.

Well besides some odd errors being thrown about rigid wheel bodies (which may be perfectly normal in 1.1) I can't see anything wrong with the log.
Oddly the dxdiag seems to feature at times a different language. But I did spot this one entry that references KSP.
Analysesymbol:
Leter etter løsning en gang til: 0
Rapport-ID: 8d94f97a-0994-11e6-83db-7824af85d9f9
Rapportstatus: 4
Hashet bucket: 
+++ WER4 +++:
Feil-bucket 129015073468, type 5
Hendelsesnavn: RADAR_PRE_LEAK_64
Svar: Ikke tilgjengelig
Cab-ID: 0

Problemsignatur:
P1: KSP_x64.exe
P2: 5.2.4.2391
P3: 10.0.10586.2.0.0
P4:
P5:
P6:
P7:
P8:
P9:
P10:

Vedlagte filer:
C:\Users\adahl\AppData\Local\Temp\RDRBAB6.tmp\empty.txt
C:\Users\adahl\AppData\Local\Temp\WERBAC6.tmp.WERInternalMetadata.xml

Disse filene er kanskje tilgjengelige her:
 
Perhaps this has something to do with your problem? Also, what you said about admin mode making it work, makes me think it could be a read/write protection problem, try moving your entire KSP folder somewhere else, like my documents. Then right click the KSP.exe, set properties, compatibility, then select run as admin so you don't have to do it each time you launch. Then right click the KSP folder itself and make sure "read only" is unchecked.

Probably not.
I'd wager the "lighting" you see has nothing to do with the "sun" you see just like every other video game.
It's all smoke and mirrors. Pay no attention to the man behind that curtain.