mk1980

it flips because the center of lift (blue/black indicator) is in front of the center of mass (yellow/black indicator) you can try to move the wings further back so the CoL ends up behind the CoM

you overlooked the "5" in front of both numbers. it's ~5760km AP and ~5180km PE. in other words your orbit is supposed to look like the orange target orbit displayed by the game. you can actually ignore the numbers in the contract description and just eyeball it based on the displayed target orbit.

the AN and DN nodes mark the intersection between your orbital plane and the plane of the requested orbit. the game only draws them on the target orbit line and not on your current orbit. if the angle between the planes is 0, you don't have to change your inclination. you put a simple prograde maneuver at your AP that raises the PE to roughly the required 5000 km and add some radial or anti radial to (roughly) line up the AP and PE with the locations of the 2 nodes on the target orbit. you usually don't have to be super accurate. the contracts are fairly generous

in my experience large planes are much easier to balance if you move the engines (and main wings) from the back to the center of the plane. basically like the concept of the real life "Skylon" SSTO project. you'll have your cockpit and passenger cabin in the front and the fuel tanks along the fuselage, with engines and wings roughly in the center. so the CoM will gradually shift a bit forward as the tanks run dry. which makes it easier to keep the plane flying straight.

if you unlocked the mammoth and the mainsail, you should also have the kickback and thumper SRBs. and probably also the recently added 2.5m monsters (haven't memorized their names yet) if you don't have those, you should probably check if your installation is broken.

depends on the goal of the design. getting kerbals to LKO is trivial at this stage in the tech tree. the only pragmatic reason why you'd even want to use an SSTO plane rather than a simple disposable rocket is to save some money. to realistically save money, the plane must be reasonably efficient in both fuel spent and playtime spent. if the mission with the plane takes 20 minutes longer than with a rocket, it's hard to justify using the plane even if the plane costs close to 0 funds - simply because a disposable rocket would get the job done for maybe 10 or 20k currency and you can make a whole lot more than that in the 20 minutes you saved by doing some other trivial contract. a reasonably fast plane with a pair of aerospikes and a pair of ramjets can hit orbit witin 5 minutes or so. landing takes a lot more time and effort than the reentry of a dumb capsule at 4x time warp of course, but if the design is somewhat stable aerodynamically and you figured out where you have to retroburn and what trajectory to aim for, you can also get a spaceplane down to the spacecenter within a few minutes. of course none of that matters if you have more fun using planes or if planes are one of the goals / constraints you set for yourself in that specific campaign. having fun trumps pragmatic effiiciency considerations. it's a game after all.

if the plane is desgined to fly to LKO with a few hundred m/s left i don't think it makes any sense to use nukes. i'd cut the nukes altogether and use 1 aerospike instead. saves about 5 tons of engine mass (which you could simply replace with 5 tons more fuel) and gives you 50% more thrust when the jets flame out (nuke has 60 kN per engine, a single aerospike has 180). when doing spaceplanes with decent amount of thrust, the most simple approach is to just aim for something like 10-15° climb and let them run hands off. no need for fancy ascent profiles - those are only really relevant for very tightly optimized configuations that have to go supersonic in level flight because they lack the thrust to accelerate beyond transsonic while climbing. you have 2 ramjets on a tiny plane, so you can just find a suitable climbing angle you don't have to change at all.

i don't think you have to make the fairing that wide. the solar panels stick out a bit and the scanner antenna is a bit larger than the 1.25m cross section, but the conical shape is impractical. try something like this maybe: https://imgur.com/a/wgQynBy pointy at the top, no sharp edges, mostly cylindrical body. as long as you don't oversteer and keep it pointing prograde you won't even need fins

you could also redesign the lander a bit so it doesn't require a 4+ meter wide fairing to contain it. you probably don't need 2 terrier engines for a minmus lander of that size. a few small radial engines would be enough, really. if the fairing isn't much wider than the 2.5m profile of the launcher rocket, some of your aerodynamic issues will already be resolved by that alone. also i think you use too many stages and too many engines and not enough fuel tanks for an efficient launcher. a skipper engine with only the mid sized (3200) fuel tank is wasteful. skippers have enough thrust to push a 6400 (jumbo) 2.5 meter tank and payload. or you use a mainsail and 2-3 jumbo tanks and a couple of kickback boosters. the 1.8m booster rockets from the first image are way to powerful. i always found it hard to aerodynamically balance unrealistic rockets. once you change it to something that resembles a real world launcher system, it will probably work just fine. i mean there's a reason why real rockets look like they look :)

an alternative to reusing stuff is to simply design reasonably efficient rockets. even in the "hard" difficulty you still get 60% of the base income. for most of the contracts the advance payment alone is still enough to fund the whole mission even at 60% and the actual payment is still pure profit. if you tweak the income down even further (custom difficulty) you will have to reuse stuff to make a profit, but at that level the game can become a grind.

you can use this online calculator to do the maths for a so called "resonant orbit": https://meyerweb.com/eric/ksp/resonant-orbits/ if you want multiple satellites on the same orbit separated by specific phase angle you'd launch them as one vessel into a resonant orbit and then detach and circularize the individual sats at the PE. simply put you basically launch them into a polar orbit of (for example) roughly 50km PE and 200km AP and then circularize the first satellite on the first pass through the PE (ie. retro burn to get its AP down to 50km) and then circulatize the 2nd satellite on the 2nd pass. the other satellite should be (close to) the opposite side at this point in time. you can launch the whole thing in one go, even combined with the tourist bus vessel that goes into an equatorial orbit. you just do the munar transit burn with the combined vessel and then decouple the satellite carrier vehicle with the 2 sats and use a tiny correction burn while you're still near kerbin to change the equatorial flyby into a polar flyby for the satellites.

my usual "algorithm" to solve that problem is to break the required maneuvers down into 2 steps. step 1: put a maneuver node on your current orbit that lies on the intersection between target orbit and current orbit. easiest way to do that is to rotate the camera to the point where you see the target orbit "from the side" and to line up the AN and DN markers. if you click on the spot between the 2, the program will place a maneuver node that is on the desired intersection line (or close enough). then set a simple prograde maneuver that raises the AP of your current orbit to the target orbit. ie. at the end of the burn the AP will be at the same spot as the AN or DN marker of the target orbit (or within reasonable tolerance). execute the maneuver. step 2: put a maneuver node at the new AP we established with the previous burn. at this point you can tweak the prograde, (anti-)normal and (anti-) radial vectors to line up the projected new orbit with the target orbit in a single burn. i'm sure there are more fuel / energy efficient ways of doing it. but it works for me. makes satellite contracts pretty trivial.

if you didn't unlock the ability to EVA yet (i think it requires some building upgrades in career?) you can use the "Experiment Storage Unit" part. it has a function to collect all data from all experiments on the vessel. IIRC the storage unit part is unlocked by the same tech that unlocks the science jr. part it's a 0.625m part so you can (for example) put it between the mk1 capsule and the parachute or put it in a 1.25m service bay below the capsule.

don't worry about your piloting skills. we've all been there :) it's easy to overdo it. you don't need much input to get a stable rocket to orbit. you just give it a gentle nudge eastwards shortly after liftoff so it falls into a gravity turn that pretty much gets it into a prograde equatorial orbit with almost no further input (other than cutting the engines when projected AP reaches orbital height and performing a circularization burn once you get close to that AP) if it's a stable desgin with fins you can usually even turn off the SAS and it will keep its prograde orientation without any input and "fall" into a gravity turn on its own. or you set it so hold prograde.

looks like a decent tutorial to me. you should be able to replicate the steps in it. i guess the thing to take away from that video is that it makes sense to use more fuel tanks on your rockets. the vessel in the video has a similar engine configuration as the rocket you posted, but it has roughly twice as many fuel tanks for each stage. which means it will have less drag issues from accelerating too fast and the center of mass will move further upwards as the lower stages burn their fuel and lose mass, making it less likely to (try to) flip over.