Often, but not always. It may stray away from the key, but it will return to it.
The key is really the tonality where the piece of music or section of music rests. Usually a piece will resolve to the tonic (I) chord at the end of the piece.
It's crucial. You may be concentrating on the chord tones, but playing passing tones between the strong beats. It will sound odd if you play a flattened 6th on the ii chord or a natural 2nd on the iii.
Yep, you should be thinking of both things at the same time.
Concentrate on the chord tones, but be aware of what non-chord passing tones are available to you. While there's a justifiable objection in thinking of each chord in a progression equating to a mode, practicing the modes with their arpeggios is a good way of internalising which notes you can use.
Just tell me this....does this ever get easier????

) and it took me a couple weeks to catch on. I would really suggest going to your local community college and taking music theory there. Taking the class opened my musical knowledge wide open. I'm taking Music Theory 2 next semester and that deals mainly with composition...so that should be really rough....