Per Google - What are the odds of getting HIV from dental instruments?
Answer. The risk is incredibly low to no risk. HIV begins to die once it leaves the body and becomes unable to infect. HIV transmission can only occur when there is a direct and prolonged exposure to body fluids, semen, vaginal fluid, blood or mother to child through breast feeding (if the mother has a detectable viral load).
The only way to know if you have HIV is a blood test. Any dentist office has safety measures in place for patient safety. They do not use the same dirty tools on multiple people or they would lose their license and be held accountable legally for making people sick. I'm an HIV Peer Educator in a hospital - the patients we see with rashes have AIDS diagnosis.
No one here can diagnose you based on a picture of a rash, there are many causes for rashes. I know a lot of people who were diagnosed and had zero symptoms at the time. The only way you will know is by going to a doctor.