About her piloting skills, one of the new books ("Rey's Survival Guide") mentioned that in the course of Rey's scavenging, she had found some computer cores and flight sims which she used to teach herself how to fly.
Shame we didn't see that in the movie. Maybe she turns on the sim before she goes to bed that time, or something.
It's explained in the opening crawl that it arose from the ashes of the Empire as soon as Luke went AWOL. The technology of the Starkiller Station is more destructive, but they're not anywhere near as powerful as the Empire, because they don't control a large section of the Galaxy, or have as much of a vast military presence. They're not noticeably at all as powerful as the Galactic Empire was, they just possessed a bigger gun.
If you dig into the additional details, the New Republic didn't even really think of them as a significant threat, which is why Leia and the Resistance didn't get along with them so well.]
How do they have the pull to built a superweapon that is ten times more powerful than all the empire's resources could, then, in secret? And how can they possibly continue the fight after losing said superweapon and most of the staff inside it? (Maybe the next movie will have them a tattered remnant of their former selves on the run, but I doubt it.)
The story would have worked so much better if it was just a big base.
Insofar as Finn killing other stormtroopers: A bunch of people around him just mercilessly gunned down a whole village of people without hesitation, and he is 100% not ok with that. I don't see any reason why he'd hesitate to shoot them down and kill them to escape. It's not like he's against killing or shooting people -- he's against what the First Order is doing specifically and if suddenly you realized that you're surrounded by the new generation of space Nazis, there's no reason to hesitate taking them out, because they're going to kill you if they find you.
Well, he seems to care about that first guy that died. I'm not saying that he can't kill other stormtroopers, but he shouldn't exactly be cheering about it, if he broke the conditioning, how does he know that others aren't fighting from the inside too?
Rey is a good pilot, but not by any means unbelievable -- especially not when compared with a 9-year-old-Anakin or A New Hope Luke. She IS a brilliant mechanic because she scraps to make her way through life, and she's a tough fighter because she lives a harsh life where he needs to be able to defend herself. Insofar as her fight with Kylo Ren: please refer to this imgur post that more adequately summarizes what I've said before.
Brilliant mechanic: perfectly fine. Not the same thing as brilliant pilot.
Luke had flight experience already on Tattoine, as well as shooting small targets at high speeds 'I could bullseye womprats back home'. He was along for the ride against the death star because they needed every pilot they had, and didn't do anything that spectacular in the dogfight before the final shot, which he only got because he was the last man standing, he wasn't the rebel's first choice to take it. And Vader had him right in his crosshairs before Han showed up out of the blue. That was a good example of force sensitivity boosting training, not replacing it.
Anakin in Phantom Menace- my memories are hazy, but wasn't that sort of an accident? He accidentally turned it on, and the autopilot took him with the rest of the fleet, or something? And yes, that was pretty stupid.
Sure, Rey is a good melee fighter, but picking up a lightsabre for the very first time and being able to put up a fight against someone trained by Luke is a bit hard to buy. He is hurt, but if they were going for 'they can put up a fight because he's hurt', that was the wrong way to frame it.
If that was what they were going for, it should have been a chase, rather than him somehow getting ahead of them, with Ren staggering along, pulling himself forward with trees, with the others doing their best to stay out of range, engaging only when they have to. Instead, they both choose to engage him.
If he's so badly hurt (and punching his wound so as not to pass out was a nice touch), Ren should be pressing his advantage as much as he can, he needs to end things quickly before he collapses. He should have punished Rey for closing her eyes for ten seconds in the middle of a swordfight.
How about this instead: Rey gets disarmed and held at swordpoint. She does the same 'drawing power from the force' thing, which Ren allows as it looks like a normal 'oh shit I'm going to die' reaction. Then she shoves him away with the force to a good distance away, and then the chasm opens up, and Ren is in no condition to jump it, so Rey goes to check on Finn. That would be more in line with 'raw power, not well trained (yet)' for Rey, without making her impossibly skilled. Neither Anakin nor Luke were able to engage in lightsabre fights in the end of the first movie they appeared in with them, the very first time they picked one up.