Welcome to the STD forum. Good questions.
Various types of HPV are more prone to infect some parts of the body more than others. The genital types rarely infect the hands, and the types that cause common warts on the hands, feet, etc rarely infect the genitals. Most likely your hand and genital wars are due to entirely different HPV strains. People rarely if ever transfer warts from genitals to hands or vise versa.
There is not complete agreement among the experts about HPV persistence. Some believe that all infections persist at some level for life, or at least that HPV DNA persists in infected tissues. Others aren't so sure. What is clear, however, is that the large majority of infections clear up to a point at which the most sensitive available HPV tests can no longer detect the virus or DNA. When that happens, transmission to partners and later recurrence are uncommon; for all practical purposes, the infection can be considered cured by the immune system. However, this is an area of uncertainty and future research could change our understanding. Stay tuned.
I hope this helps. Take care-- HHH, MD
As far as known, cuts or other open sores make no difference in risk of HPV or warts.