1) Diana brings back some medicine from Themyscira that restores her. She's major supporting cast for quite a while, and... significant beyond that as well. We'll see a fair bit more of her.
2) This is, theoretically, the same version of Diana as we see in Gail's stories now. A lot of the difference in behavior between then and now can be explained by how very young she is here, and how new to the world; a lot of it is just down to writer variance, though. Plus much of Perez' work has been retconned, a process Gail is adding to as we speak. Some has been retconned back, other wounds still remain. She's about as much the same character as Supes today is the guy in the Byrne reboot.
3) What are you, nuts? This ain't Xena fandom, here. Though if you look hard enough, Rule 34 suggests the answer is yes.