"Rise above it" can be used in a variety of situations. It means to overcome some kind of adversity or problem. Often it would be used when two people are having a conflict. "Be the bigger man" would typically be used for a situation in which two people were having a conflict. This would involve doing the right thing, e.g. forgiving the other person or making amends somehow. The sentences don't mean exactly the same thing, but their meanings are similar.

Yes - except 'be the bigger man' is sexist.

Absolutely. You can also say "Let it go".