Depends. Hpv can easily "cured" if treated appropriately. However, the virus may remain in the penile skin although not visible to the eye. These viruses may cause a recurrence of the warts in the future and close monitoring is neccessary. If no visible lesions are present, then one can be considered "cured" but must be monitored for possible recurrence in the future.
Hpv is dormant. When no lesions or warts are seen, there usually is dormant virus in the neighboring skin that is waiting for activation. You should see a dermatologist or urologist frequently to check the skin and make sure there are no new lesions trying to surface. You could spread the virus to others and back again (ping pong effect) so condoms are very important to protect you and others.
Unknown. Rectal hpv is common in gay males. Gay males need a anal pap for culture and cytology. If hpv is present anoscopy needs to be done. This does not simply go away. It is a lifelong commitment with monitoring with a good rectal surgeon or proctologist. It should not be disregarded..I have a handful of males that went on to develop anal cancer from hpv.