Self-pardon's have never been attempted, but I believe there's some dicta suggesting it would raise Constitutional issues (though it's true that it's not explicitly barred by the Constitution). There's a whole separate issue over whether or not you can actually charge a President, but it wouldn't prevent a President from being charged with a crime related to actions taken prior to assuming office (or the extreme case of a President who had been convicted prior to assuming office, although I think that may prevent running in the first place).

Impeachment isn't a criminal process. It's a political process that rests entirely within the Legislative branch. So it's not as much an exception as a wholly separate proceeding.