1) Will my wife ever be rid of HPV or will the warts continue to reappear? Is it possible to use drugs to get rid of HPV?
I can't predict the course this may take with your wife, and can only address this in general terms of what is typical, which I understand is frustrating. Long-term persistence of HPV, that is, having warts that just come back ceaselessly, is not common because, as you state, the immune system usually figures out how to clear or suppress the virus over time. HPV infections in older women (and keep in mind that in the HPV world, "older" is used beginning around age 30) can be more stubborn, as they are in people of any age who might have any issues that weaken or slow down the immune response (smoking, having certain medical conditions, taking drugs that suppress the immune system, and who knows what else).
If she's yet to do so, she might address the issue with her health care providers, asking if they recommend anything that might be of help with recurrences. While this should not be taken as a recommendation or advice, I will mention that a couple of small studies suggest that couples who share HPV and use condoms clear the virus a bit more quickly than those who don't use condoms.
2) Were the lumps on my thumb and finger connected with HPV?
Here as well there is no way I can say yes or no, but will tell you this is an unlikely scenario. The types of HPV that cause genital warts don't often cause warts outside of the anogenital or oral areas. Speaking to that last point: While oral warts are hardly an epidemic, it's probably a good idea not to have oral sex while warts are present.
I do hope this helps. Please post any additional queries you may have.