Advantage won’t help if its mites. Don’t bathe him if you can help it, it’s very stressful for a rabbit.
Ivermectin would take care of the mites if it’s mites. Rinworm is a fungus and needs anti fungal treatment, it won’t be beat with either ivermectin or advantage. I’d skip the Advantage altogethere if I were you, it’s no good against either problem, it’s for lice and fleas and may have some effect against ticks.
A bunny often get mites from it’s mother in infancy. You don’t notice them as they are in very low numbers. If the bunny later in life gets sick or stressed out for some reason, a heavy molt may be enough, the mites will get the opportunity to multiply quickly and you get an outbreak.
To me what you describe sounds like mites. Cheyletiella mites are very common in rabbits. Ringworm is common in guineapigs but not in rabbits (although it does happen). Mites are easily cured with selamectin (Revolution) or ivermectin. If you want to be sure about what is causing this, you of course need to take your bunny to a vet. We are not vets here and can’t tell you what medications you should use, you have to make these choice yourself.