Warning: rant ahead from an angry trans man who has not had the best experiences with medical professionals.
To answer the original question, if you're dealing with an adult then you deal with them in the same way you would deal with another patient. Some trans men find gyn exams extremely stressful or are too scared to go at all, others aren't so anxious but will most likely feel awkward. I found a lump on my cervix and got it checked out, I was not worried about the physical exam itself but more the reaction I would get from the doctor. Both my then GP and the doctor at the hospital dealt with it brilliantly - they made me feel at ease, discussed everything with me before doing anything and didn't ask any unrelated questions about my body or specifically about my GID. It turned out to just be a nabothian cyst so everything was fine and I would be happy to see those doctors again (though unfortunately I now live in a different area).
You would be amazed how some medical professionals treat trans people (I'm not even going to start on how they are treated by the gender identity specialists). I had a friend who got to the pap test age in the UK and booked an exam because if your gender marker is male then obviously you don't get a reminder letter sent out, the nurse accused him of lying about having a vagina (surely a quick glance at the medical records would confirm this?) and refused to do the exam without a letter from his GP - who was in the same building.
If you're a woman, imagine that every doctor you have ever seen about anything has asked you how big your vagina is and what your favourite sexual position is. Every time I see a new doctor I think oh God, what are they going to say? Are they going to treat me like a human being or interrogate me to satisfy their own curiosity? Are they going to have actually read my notes or am I going to have to explain my life story for the 200th time because otherwise they'll try to take me off my hormones because they're worried about cancer of the prostate I don't have? If I have a blood test are they going to use the correct assays for the results, and if they don't will my health suffer as a result?
Maybe it's easier to point out what NOT to do:
- You do not ask unnecessary or medically irrelevant questions about someone's gender identity or sex life (such as asking someone about the size of their clitoris during a psych evaluation).
- You do not try to look at or touch someone's chest or genitals without their permission and/or when it is not necessary (I was hospitalised for an OD and a doctor put his hands down my pants on the pretext of examining my stomach - I have a feeling that if it was for a legitimate reason he would have told me he was going to do it).
- You do not offer your unsolicited opinions on completely irrelevant topics (I had a GP who refused to give me the hormones prescribed by the hospital and told me it was because if I transitioned I'd never find a sexual partner).
- You do not insist on using the wrong pronouns (a doctor referring to me on a letter as a 'lady') - if you are not sure which pronoun to use, ask which they prefer.
There are very few circumstances in which you need to know someone's name at birth. If, say, you're having trouble finding medical records then it may be appropriate to ask what someone's previous name or birth name was, not what their 'real' name is. Don't ask just because you're curious, or start using that name if you do know it.
Be aware that sex reassignment surgery costs a lot of money and can be very difficult to fund - a lot of insurers in the US won't pay for it, the NHS may refuse to depending where you live in the UK. If you have to find the cash yourself you're talking about thousands or tens of thousands of pounds/dollars, plus the cost of appointments and travel, hormones, changing documentation, specialist clothing (e.g. chest binders, which are lycra vests which flatten your chest) and no doubt other things I've forgotten. Because of this it is not appropriate to make a judgement on someone's identity purely based on what surgery they have had. I haven't had any surgery yet because the NHS in my area refuses to fund it and because my GP believes I should just be happy with being on hormones so I don't really "need" surgery. I am no longer seeing a specialist because my father's funeral was on the same day as an appointment - I rang to rearrange it but wasn't allowed to because the specialist said that wasn't a good enough reason to miss it - and I can't get surgery without a re-referral which my GP refuses to do.
As for public toilets, I was once dragged out of a toilet cubicle, pants around my knees, in a student bar by security staff because they claimed I was using the wrong toilet and when I produced my student ID with my photo on they said it wasn't enough so I had to empty the contents of my wallet into their hands to show them my debit card, my credit card, my library card and whatever else was in there to prove that it was actually mine. I don't know what would have happened if any of that documentation had had a different name on it. In the same toilet on another occasion I was threatened by a group of 7 very drunk, very large men who also claimed I shouldn't be there. In desperation I pulled out my ID card and they claimed I'd forged it so I could use the men's toilets - an interesting concept I feel but they seemed to think it made sense at the time. Thankfully 2 friends got me out unscathed but if I had been alone I could have been in trouble. Interestingly when I was physically threatened the security staff were nowhere to be seen.