would be cool if you can rematch someone and have an option where you could pick what move you want to start at from previous game....

Agreed, I have seen a ton of games that I have done where I would like a mulligan

This idea has a lot of merit.

Good idea!  I have submitted it to our team.  Obviously the rematch would have to be unrated.  I can't make any promises, but the people that need to know about these things will be told.  :)

If both players agree, then this can be done manually even without any special provisions offered by the site.  You just do it (open two browsers, one with the old game as the template, one with the new game).  But yeah, it would be so much easier if it were automated.

Note that you can also open up an analysis window and "play it out" from the given position, with your opponent following along and suggesting moves in the chat.

 if both sides want it to be rated then whats the difference?