#1
I used to use a .60 for my sixth string and put on a .56 and it is starting to rattle when I play. It sounds perfectly fine when I play through my amp but when it is not plugged in, the sixth string rattles and I don't know if I should worry about it or not. Is it a problem?
#2
Its called fret buzz and it pretty much happens anytime you put new strings on. Let the strings get stretched out and it should stop.

If not then you need to raise your action slightly.

If it doesn't come through the amp then I really don't see what the problem is, though.
#3
Probably string stretching as said before.

Or if you have Floyd rose trem, those don't go well with different tensions