Welcome to the forum!

1) If you have a false awakening, you'll still wake up when you normally wake up. There's no way to "get stuck" in a lucid dream - that's just one of the many things Inception gets wrong about lucid dreaming.

2) I can't say I have any personal experience with intentionally improving my focus in lucid dreaming. But considering how lucid practice is all about improving your awareness, I'd say just being lucid naturally improves your focus.

3) I honestly believe everyone can lucid dream. Though the ability to naturally lucid dream without any practice is somewhat rare, being able to teach yourself to lucid dream itself is universally possible - the only requirement for you to have lucid dreams is that you have regular dreams, since lucidity is just realizing you're dreaming while you're dreaming.