They can as their secret identities.

Maybe not Superman though, I think he might be an undocumented alien.

Batman doesn't vote, he buys senators directly

They are, and even allowed to stand for office. One thing from DC that seems cool to me (from a distance) is the political conflict between different heroes, e.g Green Arrow has a reputation for being a typical Californian free-pot-for-everyone liberal. But as a character, he is also a billionaire.

Depends on the hero.
Wonder Woman can't - in the US at least; I don't know about elections on Themyscira.
Hal Jordan, John Stewart and Kyle Rainer can all vote, and they noodge Guy Gardner into voting.

Only if they have voter ID