I don’t know about private clinics but The Red Cross and Vitalant perform the procedure. And it looks like you can do it 3 times a year. Vitalant mentions “Ideal blood types include O-negative, O-positive, A-negative and B-negative.” so I don’t know if everyone can donate. They also mention high hemoglobin as a disqualifier so that might be an issue for you.
I am not sure where the blood is initially stored after removal but at a minimum it goes through plastic tubes to exit and enter. If you take my blood out just keep it. lol