[Orbit or Direct Escape Trajectory?]

I tried the two different methods from the Mun and found out that the two technics are equivalent in terms on delta-V (which makes sense).

However on Kerbin a vertical ascent requires much more delta-V. I deduced that atmospheric frictions were responsible of that.

As said above you can do a "direct ascent" with a gravity turn but that just mean you're making an orbit without stopping the engines for your escape.

For me it's orbit.

Orbit will always require less delta V than just burning straight up. Doing this test on the Mun makes this less apparent, because gravity losses are small. Try the same on Tylo, and you'll get different results, even though there's no atmo.

[Your most successful/least successful missions and what went wrong/right]

My most successful mission would be science farming Duna and Ike back in 0.25, brought along enough science experiments to get almost all of the science, returned to Kerbin with over 3k. My least successful one would be a Jool 5 attempt. Didn't realize fuel was feeding through docking ports from the landers to the transfer stage, ended up landing on Laythe with 600m/s delta V in the tanks *facepalm*, ended up scrapping that mission.

[Why Landers?]

The fuel you need to eject from the Mun to LKO/aerobreak is so small compared to landing/taking off from Mun, that the hassle to dock with it is not worth it.

Hell, if you land on the Mun's east(?) side, you can just launch straight up and never bother orbiting the Mun at all!

Launching straight up, aka radial out is inefficient, because you're fighting gravity all the way. Following prograde will always cost less delta v.

[Nuclear engines not working in 1.0!!!!!]

I've done a bit of testing with a common design I used for interplanetary transfers, an MK1-2 command pod, 3 half rockomax tanks and 4 engines, the results were as follows (using KER):

4x LV-909s: 0.40 TWR, 5.2k dV, 61 tons

4x LV-Ns (no oxidizer of course): 0.55 TWR, 5.2k dV, 44 tons

So, you get the same amount of delta V with a better TWR and less weight. Sounds good to me. When I added enough LF to approach the weight of the LV-909 setup, I had about 6.5k dV. So yes, when it comes to bigger ships with low TWR, LV-Ns are still king.

[Preferred First Rocket Stage in Contracts Mode]

Same here, I don't pay too much attention to cost effectiveness, so I go with liquid fuel, but if my lifter stage falls a bit short of achieving orbit, I'll strap on just enough SRBs to get a TWR of 1.2 with only SRBs firing, to get the first 2 or 3 km height before igniting my liquid fuel engines. I did just that when designing my Duna mission.

[Lowest Kerbin Periapsis]

One more tip, you should always be flying the ship that does the challenge. Otherwise I can set a PE just above the ground, exit to the tracking station and the ship will be put on rails, IE not slowed down by the atmosphere.

[0.24 Some feedback about contracts and funds]

To be honest, I also expected more varied missions, like:

-Launch a probe into orbit with X AP/PE and Y inclination, possibly with the addition of required parts (science, comm...)

-Unmanned missions preceding manned missions (would require slight rearranging of the science tree)

-Impactor and soft landing probes

-Rover missions

-Biome specific science missions

From a design perspective, I absolutely understand and agree with Squads decisions though, they made the system and released it for testing, fine tuning comes later.

A few other ideas, the Explore Eve mission should have a warning attached to it, a newer Kerbalnaut who just experienced the high of a Mun landing could tackle the Eve mission

and get a swift kick in the rear when he tries to bring his Kerbals home from the surface of that monster.

Test missions of the "in flight over Kerbin" could be slightly more lax and not attached to the staging controls, for instance "Use a Turbojet engine to achieve a speed over 700m/s and altitude over 20km and

recover the craft from the Runway"

Also, something not attached to the contract system, delta v stats and phase angles should be visible without mods.

[How do I complete a rescue contract?]

How close do you have to be in order to use the [ or ] key to switch to the Kerbal? I have no monopropellant on my rocket and the closest I can get is 10km.

Monopropellant is only "needed" for docking (some people dock with solid fuel ), rendezvous shouldn't be a problem. I also use a ship with no monoprop for rescue missions and don't have trouble getting within 300m of the EVA Kerbal, then I switch and bring him on board with his jetpack.