Do I have to pay for unauthorized Vet services?

Well, he still gave the shots and the wormer when your trainer said to do it, so it's in no way the vet's fault. You are still responsible for the bill. I would suggest taking it up with the trainer instead of the vet - the vet was just doing his job. If you didn't want them done, you should have been there when the vet was there if you wanted to guarantee he wouldn't give unwanted shots. Sorry for the bad news, but that's the way it works! (which is why I am ALWAYS there for any vet and shoer appointments). I would just pay the vet and be done with it if I were you, then switch boarding stables.