I prefer freezing them with liquid nitrogen, or using trichloroactic acid or podophyllin. Many doctors have purchased lasers, and other high-tech equipment, to zap these things. If not carefully controlled, these procedures may burn the tissues. Since warts are not a terrible condition, I don't think drastic treatments are warranted. Keep in mind that the serious problem is cancer of the cervix, and if your Pap smears are okay, then cool it.
All couples share their viruses sooner or later.
I assume your doctor treated you for that, and perhaps tested your partner for other STDs. The infection could have come from your present or a previous partner. Make sure that your present partner does not carry chlamydia or gonorrhea before you give up condoms. If he doesn't, and you are both HIV and hepatitis B negative, then I think you can stop using condoms, unless you continue to need them for birth control. This is one doctor who makes housecalls.