Persistent or recurrent genital warts is not a sign of HIV, except once in a while in people with advanced HIV disease, typically of several years duration. Even then, such persons virtually always are very ill and have many overt, obvious manifestations of AIDS--in other words, warts alone is rarely a sign of HIV, if ever.
1) How many men have genital warts but don't have HIV? Almost all of them, well over 99%.
2,3) The explanation for persistent or recurrent warts is not known, but a "weakened immune system" has nothing to do with it--at least not as yet discovered. There is no reason for you to be concerned about any kind of serious immune deficiency.
Hang in there with treatments your provider suggests; eventually most people in your situation stop having recurrences. If you haven't seen a dermatologist for a second opinion, I suggest that as well. Among other things, after numerous treatments, scar tissue resulting from treatment sometimes is confused with recurrent warts; a dermatologist can sort this out and also might have other treatment advice. In the meantime, you should look at this as an unpleasant inconvenience, but not a serious health risk or indication of any important underlying health problem. Assuming you have no other HIV risks you don't describe, you can be confident your upcoming HIV test will be negative.
Good luck-- HHH, MD
We all got so lucky that HIV is as hard to get as it is. If it were as contagious as HPV or Herpes, half the people I know would have it.