I would agree with Two and Anonymouse, but with a caveat, that River isn't the pilot in It's Never Easy. That could be because she's playing with the horsie, or because the lucidity she had when she saved Mal diminished, maybe she's grounded after she did something Mal got annoyed at. But she doesn't appear to be 100% reliable, which is either due to being a girl just out of her teens, or it could be her continuing mental state, or both.

I agree with TWO & ANONYMOUSE thanks 4 the comments!

I wouldn't let a crazy person repair my car or drive it. I expect the same from Mal about his spaceship.

River is competent to run tests on the control wiring above the engine room ceiling and Mal does not lock the pilot controls on River's side of the bridge. (There is a pair of pushbuttons at Mal's position to do just that. Mal takes control after liftoff.) Mal trusts River.

Maybe it also counts for judging her mental competence that River saves Mal's life in the free comic that takes place months after the movie: Serenity: Firefly Class 03-K64—It’s Never Easy. http://www.fireflyfans.net/mthread.aspx?bid=4&tid=51833

She could never be completely "fixed", not with that kind of abuse to her brain matter. But it's beyond question that she's better. The defining moment, I thought, came the second time she said "I'm all right", just after she threw up. The way she said it gave the very firm impression that she wasn't just saying it, she was realising it. Up until that point, even she believed she was crazy, not knowing what was real and what wasn't; the confirmation that the events on Miranda weren't just her imagination or aberration, but were in fact true, brought her a long way back towards sanity. I still think it was some of the best acting I've ever seen from any actor; Summer's always good whatever she does, but in that scene she totally outdid herself.

Yea that is one way 2 look @ it.

Not completely whole, I don't think. It is likely she has other secrets hidden in there, and there is still the problem with what physical alterations they made to her brain. With no amygdala she still has no control over her emotions.