3) Genital warts only appear in the genitals or mouth. You wont spread them to other areas like the hands, as the warts found on the hands are from an entirely different strain. Even if you did touch your genitals and then your mouth, it'd be so unlikely for you to give yourself oral warts because as coljack said above, the virus dies pretty much within a second of being on your hand. Also oral warts are rare, and are mainly transmitted through oral sex. Also if the warts have been treated then you are less contagious down below.
4) Don't irritate the area. If you want to speed up the healing process then take vitamins specifically for the immune system, eat healthily, stop drinking/smoking and generally stay healthy
First off, it is likely that you did contract genital warts, even though your partner had his warts covered, your entire genital region is contagious (the boxers area) and so it is likely that the warts were genital warts. You have had them frozen off and hopefully they will not reoccure, but if they do it most likely be in the next 6 months.
1) It is not a life long problem, a doctor on this site said that warts do not persist for a lifetime and usually the problem is solved if no warts reoccur in 6 months.
2) You do not 'ping pong' the virus back and forth from you to your partner, however on the UK NHS website they do recommend abstraining from sex untill the warts gone and use a condom for the first 3 months as sex can cause abbrasions which could slow down the healing process.
3) I am unsure of this question but it is unlikely to pass a wart to another area of your body as the virus dies almost instantly when it is away from the genital region.
4) Only do what the doctor has told you to do. There are alot of home made remedys on the internet but these usually do not work and can cause scarring. If they did work then you doctor would ask you to do it. Usually after freezing they can ask you to use a cream to try and produce healthy cells to replace the dead ones.