Welcome to the STD forum.
These are excellent questions. However, they have been asked and answered many times on this forum. Here are two threads to get you started. You can find others by using the search link and entering such terms as "HPV clearance", "genital warts", or just "HPV".
http://www.medhelp.org/posts/STDs/concerned/show/980849
http://www.medhelp.org/posts/STDs/Clarification-of-HPV-clearance/show/249312
The bottom line is that some experts believe HPV DNA persists in the genital area in most people, perhaps for life. Others disagree. What is certain is that in the large majority of infected people, HPV clears up within several months, to a point that it cannot be detected and partners cannot be infected; and in most people it never recurs. So for practical purposes, the natural course is one of spontaneous cure, whether or not undetectable DNA persists in skin cells. The wart-causing HPV types clear up more often than the other types that are associated with cervical cancer.
In people with genital warts, the HPV infection typically involves genital skin more widely than the warts themselves. Therefore, the infection may continue a while even after warts are successfully treated by freezing or the other available treatments. Exactly how long it takes to be confident the infection is truly gone isn't known. My advice to patients is that if the warts clear up with treatment and there are no recurrences after 6 months, they can be reasonably confident the infection is gone and cannot be transmitted. But it might not take that long, and once in a while maybe 6 months is too short. However, I have never seen a patient who apparently transmitted warts to a new partner in this situation; if it happens, it is uncommon.
I hope this helps. Search the forum for literally hundreds of other discussions of these points.
Regards-- HHH, MD