Superfluous J

The contract wants an Atmospheric Survey. The Science Jr (and Jr Jr) do a... Environmental Survey? I don't recall.
The Atmospheric survey is from another part a bit further down the tech tree.

I can't speak on fitting one in a cargo bay because I don't tend to use those. If I wanted to take a rover somewhere I'd probably put it under a fairing.
Rovers currently have one practical in-game use: getting closer to an anomaly when you can't land close to it. Even then, if you landed with Kerbals they could just walk/fly to the anomaly if you really need to.

Your root part is on that dropped stage.
In the VAB, click the "reroot" tool (I think you can hit "4" as well) and then click your command pod to set it as the root part. You may need to click a second time for no good reason but just do it if it stays in "root" mode.
Also, this is an indicator that you may be building your rockets backwards, from the bottom-up.
If you're building an orbiter that's supposed to return to Kerbin, you should build it in this order, attaching the new stuff to the bottom of what you already have:
The capsule, with parachutes and maybe a heat shield (not necessary right now) A small tank of fuel and an engine to do orbital maneuvers. A larger tank of fuel and larger engine to get into orbit from the air. An even larger tank of fuel and engine to get off the pad, perhaps with some side boosters.

From what I can tell, no.
You need to have at least one command part (probe core, command module, cockpit) that was on a vessel that was landed on the Mun, recovered from landed at Kerbin, to get this science.
Science doesn't count. Kerbals don't count. the science CONTAINER doesn't count. It must be a command part.
If you're lucky, you brought a probe core to the surface and you can EVA Construct it onto your lander. If not, you're either having a fun ride home or you're getting that science on the next mission.

How do you build the rocket? You are supposed to build it from the capsule down, not from the engines up.
If you're building a Saturn 5, you build the capsule that will splash down, then the part that will land on the Mun and come home, then the part that will get that to Mun, then the launcher stages. You don't start with the big bottom stage and work up.
If that's not the problem, I'm not sure what could be.

Yes.
Note that if you have "fuel transfer obeys something or other I forget the exact name of the setting" you could claw to something that would not allow transfer, but it would be that thing (decopulers and many structural parts have this property) and not the claw specifically.
And this should probably be in gameplay help but I don't mind answering now.
EDIT: A rare double-ninja! I got beaten out by 2 different posts above mine that each addressed one of my two points.

In case that picture (which you should not just bookmark, download, and print but also memorize) is not self explanatory enough.
To get to orbit from Kerbin you need 3400m/s of dV. To eject to Duna you need 950+130 or 1080m/s of dV. You will also need up to 25m/s to correct your orbit's tilt on the way. To slow down into orbit at Ike you need 250+30+180 or 460m/s of dV To land at Ike you will need 390m/s of dV To take back off you will need another 390m/s of dV To eject back to Kerbin from Ike you will need 180+30+250+130 m/s of dV You can in theory aerobrake back at Kerbin but any extra fuel should be used to slow down first. These numbers could grow based on your ability but generally won't shrink.

I'd need to see a picture to know exactly what you're talking about, but it's likely that Kerbin is rotating under you and your ship was sent straight up. It's just an illusion that you're going "backwards". In fact you're standing still (floating still?) in space and Kerbin is rotating under you.
Even if you retain the 250 or whatever m/s of Kerbin's surface rotation, it is moving 250m/s around its relatively smaller circle, while the 250m/s you retained is around a much larger circle up in space. When it's rotated halfway around, you've gone 10% of your orbit.

FIRST NOTE: These numbers are suggestions and will vary in reality based on your skill, rocket, and how you do each burn. But given a typical rocket and correct burns, they should be pretty close. I like to keep a 10% margin for error in my tank, especially for landing on airless worlds.
You need to think of the map in sections. There's a "get into orbit or land from orbit" section of flight, a "circularize orbit from interplanetary or go from circular orbit to interplanetary" section. Each trip includes one or more of these and they're not all together.
Let's take a simple one, getting to Mun.  First you get into orbit (3400m/s, which I'll skip from now on and assume we're in orbit of Kerbin to start) and then there's the "go to Mun from circular orbit" which is 860. When you're done with that burn, you should have a periapsis around Mun at around 14km. You won't, but a tiny mid-course correction should do it. Then when you're at that Pe you need to do the "circularize at mun" burn which is 280. Then to land from there you need 580.
Reversing that, you need 580 to orbit Mun, 280 to eject back to kerbin, 860 to brake at Kerbin back into a circular orbit, and 3400 to land back at Kerbin. However, those last 2 can utilize aerobraking so you can do them all for free, assuming you can stand the heat
Okay now for the harder part (don't worry it's not much harder), getting to Duna. Duna has more numbers than Mun after the initial 3400 to reach orbit. you've got 930, 130, a little '10' off of that 130, 250, 360, and finally 1450. Whew, what to do?
Remember, there are only 2 actual ways to spend fuel: landing/takoff, or circularize/eject. So, we're in orbit of Kerbin and we want to eject to Duna, so we just add up all the numbers to achieve that. Note the little flyby icon in the Mun trip we did. We want to add all the numbers to Duna that reach that same flyby icon. That's 930 + 130 or 1060. You do that burn (at the right time and in the right direction) and your ship will encounter Duna, or at least be really close, and you may need up to 10m/s mid-course to fix your angle. You also will need a tiny bit to tweak your flyby so your periapsis is 60km away.
Now you're at that Pe and you want to circularize, so you need to burn 250+360 to do that. if all you want to do is get into ANY orbit (say, you're going to Ike instead of Duna) you'd need to at least burn 250. But to land on Duna, you need an extra 360. You do those at once though so need 610, but if you're daring you can use Duna's atmosphere to do them for you.
Then to land you need 1450, though really you can do it for almost nothing and let the air slow you down most of the way. Returning, though, you need 1450 to get back into orbit, 610 to eject back to Kerbin, 1060 to slow down at Kerbin back into a circular orbit, and 3400 to land at kerbin. As before, those last 2 can be reduced to almost 0 via aerobraking.
And that's basically it, all other planets are similar to Duna.

B will brake while you're holding it.
In the top of the screen to the right of the Altimeter are 3 buttons, the bottom one is the brake and it will then stay on until you click it again, or tap B once.

Yep. So long as you set your preferences but don't otherwise play in Steam it works perfectly.

Pretty much what happens anywhere else. You get a little message and a science report with some points that you can transmit, bring home, or stuff in a lab.

Without comms, an Engineer plus a probe core can do everything a pilot can do, except make and delete maneuver nodes. The probe gives you the ability to turn on/off SAS, while the Kerbals give you the ability to aim any direction you want.
I do this all the time sending and engineer + scientist down to the surface to lay out the Breaking Ground science doodads.

Or a docking port, yes.

You don't need to flyby Jool. You just need to burn until your "orbit" around Sun is not an ellipse anymore, and then turn around and with VERY light taps, burn back until it's an ellipse. Once your Ap is past Jool it takes less and less dV (at your Sun Pe which is hopefully also in LKO).
If you DO want to do a flyby of Jool then yes lower is better, and if the encounter itself doesn't give you a hyperbolic Sun orbit then you should do the burn at Jool's Pe. But again, that is 100% not necessary and may even cost more fuel for all I know.
Note, though, that - mathematically speaking - the "highest possible" Ap over Sun is infinity. Practically speaking you'll at some point not be able to expend a low enough puff of exhaust to not send your Ap from "really high but not yet infinite" to "oops now we're hyperbolic again"

Is your Kerbal an Engineer?
EDIT:
As the Correct Answer, I should probably elaborate.
Only an Engineer Kerbal can evoke EVA Construction Mode with the I key (default).

Close by, [ and] do it in stock.
The only mod I can think of is Targetron, but I don't use it so can't really speak on it.

Exactly. Also, at the right time and in the right place.

If you like mods (and you should) you could try
 

All these mod and external map suggestions and not one mention that with an OKTO2 and Kerbnet you can see the biomes in the stock game.

One caveat, you sometimes have to be ridiculously close to the Science Jr (In particular) to take the data. They upped the range but there have been times when I had to stick my kerbal's head inside that thing to get the button in the part action window.

If you put your mouse up in the top middle of the screen a filter will drop down. You can turn on and off each vessel type. Included in "vessel type" is unknown objects, aka asteroids (and possibly soon to also include comets).

When you're higher up (and especially when you're at the top of an elliptical orbit) you are moving slower.
To turn your orbit, you need to basically come to a complete stop, and then do that same burn at a 90 degree angle. Actually it's 2 of that (not 2 funds, I'm using that as a square root symbol) because you can do it all in one burn and cut the diagonal. In total, it costs about 1.4 times your orbital speed to shift your direction 90 degrees.
If you're in LKO, you're going 2300 or so m/s so it would cost you about 3250 m/s to turn your orbit 90 degrees.
If you're up at the height of an ellipse well past Minmus' orbit, you may only be going 10-20m/s, so your turn will only cost 15-30m/s or so to turn.
It costs about 950m/s to get that elliptical orbit, and 950m/s to go back down, so instead of burning 3250m/s to shift your orbit 90 degrees you "only" have to burn 950+30+950=1930m/s. That's a savings of 1320m/s.

No it was you you listed them here.
They also told you what we'd need to help you there.
But the short of it is, KSP doesn't take that long for anybody else so the problem is not with KSP in general but with your install of it. Therefore while with good information we may be able to help, the problem can only be solved by you.

The nodes of a planet or craft is where your orbital plane crosses theirs and actually doesn't match the planet's equator inherently.
Most planets in KSP have low orbital tilt, so the equator does match the orbital plane in many cases, but Minmus is not one of those cases.
To get a perfectly equatorial orbit in the stock game, I can only think of 3 ways:
Cheat something into an equatorial orbit of Minmus. Accept a contract to put a satellite into an equatorial orbit of Minmus. Accept a contract to rescue someone or something from an equatorial orbit of Minmus. Any of those will give you a reference orbit you can then match with your craft that you want to be equatorial.
There is another option: eyeball it and accept that if you can't tell the difference it doesn't matter