I think PaPiro's answer is entirely correct, based on my own experiences as author, referee and editor.

One principle is that the referee is carrying out a task at the request of, and for, the editor. I would say that the the referee's task is to supply information to the editor, who will use this to decide what should be done with the paper. The standard is that the referees are anonymous, and if you agree to referee a paper you are accepting this constraint by default.

All correspondence between the referee and author should go through the editor. This provides the referee with protection from the author - I recall one case where an author was very vexed by a positive report, because it was felt that the report was not positive enough. It also provides the author with protection from the referee's requests, which are not always guided by the pure light of reason.

If the referee provides a substantive improvement to the paper, the authors are free to propose via the editor that the referee be added as an author. They are also free to decline. My own view in this case is that the editor does not have a right to insist that the improvements be added, and that the final decision should be based on the submitted paper, not on what it might have been. (Dealing with this sort of issue is why editors are so well paid, of course.)

If as referee you are unhappy with the way an editor has handled a paper, you are free to contact other members of the board.

As for Will Jagy's anecdote about Wiles, I appeal to the adage "extreme cases make bad law".