First Bran battle

could you just let him kill you and wait the time it takes for the mission to complete its self, or do you automatically fail the mission if you die?

I think so, but most people prefer not to lose experience... But it is worth testing and mentioning.

Also, every attack does 1 damage. So at a semi-impossible chance, is it possible to kill him? If so, what happens?

Unless you can reduce some of his protection, then yes you can kill him, ending the mission earlier than intended. and getting killed won't kick you out of the mission. I think you just sit there until it ends.