[Farael]

I really don't think that Gandalf would have died as a man, given that he was an Incarnate Maiar. Had he been a Maiar turned into a man perhaps, but from my understanding his body was sort of the 'container' for his Maiar spirit. We see him die and be re-incarnated which is not what would happen to a Human so I'd think he is still Immortal. Furthermore, I think he's "more" immortal than Elves, who were not really immortal but rather will not die of natural causes. If I'm not mistaken, Maiar are immortal as long as they are good in the face of Eru (which would explain why Gandalf was re-incarnated while the Balrog was not... and it'd also explain how Sauron outright disapeared. Saurman's is a bit different as he is shunned form Valinor and then disapears, and is sort of the exception to my "rule" but the others we know not where they go, so I'd guess that somehow Eru (the only one strong enough to actually KILL them) is involved)

Whether or not he decided to keep his Gandalf form is up to each of us to imagine I guess... I like to think that he would, given that he has experienced a lot of things in such form... at least until Frodo and Bilbo pass away. Of course, he'd live on eternally so maybe at some point he'd become more Maiar and less Gandalf yet Id think he'd stay Gandalf for a couple hundred years at the very least.