I'm in Medical School so I will take a shot at your questions, but I will put out there I am in the closet about being trans. If you feel strongly about talking in your personal statement about GID, put it in there
As I obviously didn't, still being in the closet and all, my best advice would be that if you do frame it into how it would relate to you being a Physician. For example, if I theoretically wrote my application out and proud, I would say something along the lines of, "After going through my own period of struggle, exploration, and discovery regarding my own gender identity, I gained a better understanding of underserved populations. No one dreams when as a little kid of growing up and experiencing hardship, and as a physician I could apply this facet of my own identity to heal with empathy and understanding." This being said, you have to realize your audience, and understand that there are a significant portion of bigoted Doctors out there. I would say though that if you have a passion for trans-issues, there are those opportunities. For example, a girl in my class did a poster/ anatomy dissection presentation, of her work with GRS. I wish I could have told her my true identity when I walked by and she explained it to me; she wants to become an SRS surgeon, which I was impressed by.
In response to Flan... Don't worry about Simulated Patient Encounter Videos, you learn them in medical school with much better context than you would if on your own before school. As far as the interview process, after multiple years of applying, was connecting with the interviewers and finding what they wanted to talk about; basically showing them you are a normal person who can interact with others. About your reference to GPA and MCAT numbers, I seriously recommend doing as best as you can. Yes, references and volunteer history are good, it gives the interviewers something to talk about and shows you have a tad bit of insight in to the medical system. But ultimately, this was the process at every school I had ever interviewed at, GPA/MCAT/Personal Interview/ Experiences gets you a score that ranks you on a list, and they choose from the top of this list. A fair percentage of that score is GPA/ MCAT (My experience about half)... Do as well as you possibly can, and it is one less thing to worry about

Also, when you do get in, you will be prepared for the massive cramming you have to do in school/ boards.
Any help you need, just let me know

Allison