She cannot be "convicted" of anything so long as she's in office. She would have to be impeached and removed from office (or resign) before she could actually stand trial for whatever crime may have been committed.
I have no idea if the emails are related enough, as I don't have voting rights in the US I see no reason to care about them.
The Senate is only required to actually convict. It's pretty much a disaster simply to be impeached at all. No conviction is really necessary to permanently damage someone politically.
We've only got one recent example of a President getting impeached, and it worked out pretty well for him.