It may just be me, but I keep asking myself did they even ask you if you were ok with a trainee? I would be highly irritated if they did not ask me if I was ok with this. I hate blood drawn like will get light headed so if the person doing it misses and has to go at it again, I feel so sick. I would have had a sh!t fit if I found out it was a trainee even if she did manage to hit the vein. I mean obviously she was surprised she did with her mess she made afterwards.
Also the no gloves thing makes me want to puke. The nurse is dealing with someone else's fluids. Come on, protect yourself lady. Not to mention protect your patient from your germs!
I would totally do what pp have said and at least contact the office manager. I know in my OB's office while they have a lab onsite, it is run by an actual lab and not my doctors office. That employee is not a direct employee of my OB but the lab so make sure you find out if that is a direct employee of the office or a lab that just "rents" space from your doctor. If they are an employee of the Lab then also file a complaint with them.
On the HIV test, it's a simple enough test and if you want you can have it taken at another lab. I would totally do it just for peace of mind and for peace of mind of the trainee. It wasn't completely her fault what happened and more the person training her. The only thing I would say is make sure the lab or office is paying for the test and you do not get a bill for it.