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Your story is a common one. HPV is unpredictable. Most HPV infections are believed to be cured entirely by the immune system, but HPV DNA may persist -- and recent research indicates DNA persists more commonly that previously believed. Up to half of all adults in the US, including people over age 40 and not at current STD risk, have HPV DNA detected on the penis, vulva, vagina, labia, etc. There is controversy about whether HPV testing should be done at all in your situation; it is clear that testing should be done if a pap smear is abnormal, but otherwise might not be wise if the pap is normal, especially at your age. This isn't meant to be critical of your doctor for doing the test: she is following common guidelines. But with the new research about how extremely more common HPV is than previously realized, I imagine the guidelines will change as time passes.
Despite the very high prevalence of HPV, new HPV infection with symptoms (e.g. warts) are very uncommon after age 30 or so. Even with carriage of DNA, people might be immune to developing active infection that causes growth of warts (and perhaps of precancerous tissue as well).
To your specific questions:
1,3) Most HPV DNA tests check for only a few types, i.e. the ones most likely to progress to cancer (HPV 16, 18, and a few others), but many also check routinely for HPV 6 and 11, the two types that cause most warts. You'll have to check with your doctor's office or the lab about exactly what types are detectable on the test that was done. But even if you have a wart causing type, it's not likely to actually cause warts if you or your partner don't have them already.
2) Since you and your partner apparently have been together for some time, he has been repeatedly exposed to the HPV types you have and undoubtedly carries them himself. He could have been the source of your infection, or vice versa -- you'll never know. In any case, it's too late for changing sexual practices, which would not prevent either of you from any significant health outcomes. This would be closing the barn door long after the horse is galloping over distant fields.
4) Nothing is known to speed (or delay) clearance of HPV or prevent progression, e.g. toward cancer. Well one thing: don't smoke! Tobacco use increases the frequency and rate of progression toward cancer. However, the vast majority of HPV infections never lead to cancer. When it happens, it takes many years -- during which it is always possible to detect early changes and give curative treatment before it becomes dangerous. I imagine your doctor will advise you about follow-up pap smears and testing. Follow that advice, then forget about it.
5) Penile cancer is much rarer than cervical cancer in women. Here too, any cancer he would develop would first show up as a minor bump or sore on the penis, at a point of easy treatment. Invasive penile cancer requring drastic treatment (e.g. amputation) is exceedingly rare. This also is not a serious worry.
6) Add throat cancer and anal cancer to the list. Otherwise no known ill effects. And these cancers are rarer still. And again I will stress that his risk for these is no higher on account of your HPV than it would have been if you had never been sex partners with one another. Exposure to any particular infected person does not materially raise the risk of bad outcomes.
I hope this information is helpful. Please let me know if anything isn't clear.