Does it hurt? Oh wow, yes it does. Especially the upper and lower lip, but numbing cream can help. Not every hair hurts as much as the other, some barely at all, some excruciating but most are midway. Some of it will depend on the skill and experience (number of hours the tech has worked).
What happens? Depends on the method - there is thermolysis, galvanic and blend (bit of both). I can really only comment on galvanic as that is what I use. You shouldn't shave for at least two days, the tech needs to have long enough hair to be able to see the follicles and remove the hair. You lie down on your back. The tech sterilises your face. I have my eyes covered, they use rather bright lights so the covering keeps that out. And away they go... galvanic typically is a multi probe process. I have 32 probes, one probe is inserted into follicle (that's usually the bit that hurts) and left in there for about a minute. You need to be holding onto a metal thingy that helps close the circuit, it kills the follicle by alkalising the root, the probe is removed and hair is then plucked out (painless). Then it's onto the next set of 32. My tech usually starts with 32, removes and reinserts the first 16, then removes and reinserts the next 16, etc. it's a long tedious process, but it is effective and hairs do not grow back.
How long? For galvanic it can be between 150-200 hours depending on how thick and extensive the beard is. I'm hoping mine will be about 130, have done about 85 hours, started over a year ago.
How often? The skin needs time to heal. Typically you shouldn't shave for two to three days afterwards to avoid scarring (by knocking off the micro scabs) or infection. It's good to use alcohol swabs on your face for a couple of days afterwards. So you could do them every week if you were keen. I've generally done them every two weeks but sometimes take a break for a month or more. Sessions can range from one to three hours. Generally they should start with one hour sessions so you can get used to it. Longer sessions are more cost effective. I've done two three hour sessions on the same day, that was quite a marathon. Yeesh, only did it twice. The first five hours were easy compared to the sixth! Usually I chat with the tech the whole time, really helps the time to fly!
Discrimination? I can't imagine you'd have trouble finding a trans friendly clinic in NYC. Use Google. I'd look for a place that does electro as part of their core business not just some corner beautician. The latter may say they've had years of experience, but that might only be a few hours a month on old ladies wanting a few hairs in their mo removed. My electro in Sydney does it all day, every day, all year. You should be asking them what method they use and how much experience in hours their techs have had.
Cost? Mucho dollars. I don't know about the US but I pay about $450AU for three hours. Yep, it really hurts the purse too!
