I noticed that dentist's latex gloves gives me facial rashes, but I tested negative for a latex allergy by an allergist. Can I get the HPV vaccine?

Get retested. If you notice a rash when your dentist uses latex gloves, the easiest solution is to have him wear non latex gloves (I am sure he has these available for his latex allergic patients). The other is to get retested if you want to be certain. Sometimes allergy tests can have "false negatives". And yes, you can get the HPV vaccine!