Reader query: transitioning and applying to grad school

A reader writes:

I’m a transgender woman who is in both the process of transitioning and applying to graduate school in philosophy. I have a few questions that I hope you could pass on to your readers so that I can handle my situation as well as possible. My situation is that of someone finishing their degree in one gender, starting their PhD in another, and hoping to move between all of that as smoothly as possible.

(1) I don’t want my time at my PhD school to be contaminated by my old ID, how can I work with the admissions committee or whoever in order to make sure that my status as a trans person is not shared and that information regarding the ID I applied with is kept under wraps?

(2) How can I communicate to graduate schools that I am transgender in order to make sure that they know what to expect and to make sure that that won’t be an issue for their department (and me)?

(3) I’d like to come out to my own department (or at least to people within it) in order to enlist their help with my applications, but I’m terrified of taking a wrong step and tossing all of my letters out the window. Any tips on coming out discreetly and carefully within my current department?