He absolutely does have access to your account, yes.
Tweets are stored in their database. They do have the technical capacity of modifying them.

Again, I don't think they would. There might be (in fact, I think there would be) legal repercussions if they did.
But, from a purely *technical* point of view, they absolutely can.

This isn't a matter of opinion. It is how it is. It absolutely is exactly how it works.