Yes. It is just what the person who represents the State is called in criminal matters. So if you're on trial the guy trying to get you convicted is the prosecutor.
Yes, it is quite likely that - outside of work - they probably may know one another. It is not improper as long as they both perform their duties according to the canons of ethics of The Bar and the...
If he needs one. There's a famous legal aphorism that "a lawyer who represents himself has a fool for a client." (Meaning: it's foolish to represent yourself; hire another lawyer to represent you.)