1. In the show the captain, after rescuing Simon & River from being burned at the stake, tells Simon that he's part of his crew, and he'd have done the same for any of them. In the movie, the captain says that River and Simon are not a part of his crew.
No, I think what Mal said was that they are part of his crew, but only so long as Mal says they are. Which does seem to contradict the sentiment, but people's sentiments and attitudes can change. As others said above, Mal's in a darker place in the movie. Personally, I don't see that as a contradiction, because he'd lost Inara, whom he secretly loved, and Book, who kept him morally grounded. Without their mollifying, civilizing influences, Mal lost his way.
2. In the TV show, the doctor says that he paid some guys to get River out of the treatment facility via cryo. In the movie, he rescues her himself (which I found completely unbelieveable).
True, what Simon says
in the show contradicts what we saw in the movie, but people don't always tell the whole truth. Maybe Simon didn't want to admit that he'd broken her out himself. Maybe he thought that Mal would be less likely to keep Simon onboard if he knew that Simon was capable and devious enough to stage a successful commando raid on an Alliance intelligence facility. So he downplayed his role in River's liberation to make himself seem less threatening. (Though if he'd known Mal better, he might've done the opposite, since Mal would've admired anyone who could pull off such a feat against the Alliance.)
As for Better Days
, it's clearly set sometime before Those Left Behind
, since Inara and Book are still aboard -- maybe it's even before "Heart of Gold," since there's no mention of Inara wanting to leave. Although since it features the hover-Mule, it would have to be after "War Stories," since I think that was where the original wheeled Mule got wrecked.
My problem with it is the source of their newfound wealth. What I think they should've done was to tie it into "Trash" -- have them get rich by selling the Lassiter pistol they stole. That would've tied up that dangling loose end from the series.