Does the Animagus form have any relation to the Patronus form?

Answer 1

We do not know. Professor McGonagall's Animagus form is a cat, as is her Patronus (we see that near the end of book 7). However, of the Animagi we know about (James, Wormtail, Sirius, McGonagall, Skeeter), McGonagall is the only one whose Patronus we see. However J.K. Rowling told us that James' patronus is also a Stag.

We do learn that a Patronus can change, as Tonks' is reported to do, as a result of life events, but we are led to believe, by the requirements of registration at the Ministry, that the Animagus form does not.

It can only be a guess, but it seems likely that both the Animagus form and the Patronus form are set by the person's personality at the time of casting the spell, and the Patronus can change as the person changes, but the Animagus spell, that creates the Animagus, is cast only once. So: If Harry, in his third year, had been skillful enough to cast the Animagus spell, we can guess that his Animagus would be a stag; however, after he settled down with Ginny, his Patronus could change to be something more fiery, like a fox or a dragon, but his Animagus would remain unchanged, having a relationship to who he had been rather than who he had become.