1. Oh, I forgot about asexuality. Okay, so, straight, gay, bi, and asexual. Four roads. That's plenty of choices. You don't need any more.
2. Just because someone believes they are something, it doesn't mean they are that. I could believe I'm actually an anthropomorphic fox's soul stuck in a human body, but that doesn't make it true. Besides, GID is listed in the DSM-IV, and the DSM-V will list Gender Dysphoria. Believe it or not, this can be beneficial for transgenders, as it gives them protection from discrimination, can cover gender-reassignment surgeries (insurance-wise), among other things. In short, they're best off with it classified as a mental illness, because it grants many benefits that most lobbyists overlook. Here's a document on the nature of GID in the DSM-IV: http://www.aclu.org/files/images/asset_upload_file155_30369.pdf
3. People, in a sense, do. Say I have lesbian thoughts. That's all fine and dandy. But whether or not I choose to acknowledge those thoughts is up to me. I can choose asexuality, the magical fourth road you pointed out. And lo, I have made a decision not to like that gender.