Hi Dylan,
I'm not sure what state you are in as that changes what can be done. You could get the apartment and then change your name. Depending on where you are, the gender marker may be able to be changed simply with a therapist note. The DMV tends to be much easier to change than any other document.
Some places give you protections based on trans status, some protect you so they don't chance a problem with the general sex discrimination laws that apply everywhere.
As to medicaid covering trans stuff, that is not guaranteed except in certain states. Only medicare can not flatly deny you.