What would happen to an adventurer's personal identity when turning into a God?

Question

I recently played RP with some friends, and one these friends is also one who's playing in the D&D campaign I DM (Lost Mine Of Phandelver).

While they were explaining me their home-made RP system, we came to talk about Gods.
After a bit of talking, my friend who is also playing D&D told me that he might someday try to achieve becoming a God.
It could be a logical way for his character, an elven wizard, as his goal is to be the most powerful magic caster that ever existed (The entire background may be a bit long to write here not not that relevant for this question).

I was thinking that he might someday become the new God of magic in place of Mystra.
I already know that a legendary adventurer becoming a God already happened, but it brings me to a few questions about what would happen :

1. Would the elven wizard keep his whole personality or would it be deeply affected by the previous God of magic memory and personality?
2. Will the new God be a God (male) or a Goddess as the previous one?

Answer 1

They start with their own personality, yet they can and do change afterwards.

This question is rather difficult to answer in a manner that would satisfy everyone: it is best left to individual interpretations based on precedents. I suggest you have a look at the list of deities who were previously mortals and pick some examples and read about them on the FR wiki.

If you want to focus on a couple of well-documented examples, I would suggest you read on Mystra, who has had at least three incarnations. The mortals who were chosen to carry the mantle by each previous incarnation carried some of their own personality with them, as evidenced by the change in alignment.

However it is also true that the personality can and does change to adapt to the job. Kelemvor, the mortal lover of Midnight who became Mystra, was an unusual god of the dead right after his ascension. For example, he was "lenient" on the faithless and the false. Yet this created imbalance in the world and as time went on he readjusted himself for his job. Quoting from the FR wiki article:

Kelemvor wondered how he could judge the damned, when he himself failed his own personal judgment. He gradually came to realize that there was nothing human in being a god. To correct his mistakes, great changes were undergone in his realm, as well as in himself. ... [Kelemvor] rid himself of all signs of humanity in order to properly fulfill his duties, ...

Kelemvor and Midnight's ascension and the ensuing changes in personality are described in the Avatar Series of novels, in particular the last novel of the series Crucible: The Trial of Cyric the Mad. Another ascended deity whose story is described in detail is Finder Wyvernspur - his mortal life is recounted in the Finder's Stone trilogy, while he is a deity in the novels Finder's Bane and Tymora's Luck.