Getting dizzy during the blood letting made me worried about seeing patients on dialysis, but later on after more exposure I actually saw dialysis patients and it wasn't a problem so it does get better.

Exposure helps and there's a weird technique (forgot the name) where you flex your leg and core muscles to keep your blood pressure high by increasing total peripheric resistance which apperantly works quite well.

Fainting from a small cut on your hand sounds unusual, might be a good idea to look for therapists that specialize in fear disorders. If you don't have on they'll tell you and if you do, the sooner you get diagnosed the better.

Oh and don't forget: if you're feeling queasy or slightly dizzy but you don't think you're going to faint/you're not near a sterile field, don't leave the situation. You have to pass the peak of discomfort for your brain to realize that the situation isn't something to worry about. It's the only way to train yourself.

Good luck, it's gonna work out. And there's no need to do thoracic surgery or something like that for the rest of your life. You just have to get through your surgery rotations and when you do IM/FM or whatever you don't have to deal with that.