Welcome to the forum.
First, I'm assuming your diagnosis is correct. Probably it is, especially if your doctor is a dermatologist. However, not all genital bumps are warts, and many kinds of non-wart lesions would also clear up with freezing plus imiquimod (Aldara). If there is any doubt, discuss it with your doctor.
Assuming you indeed have warts, let's consider the possible origins.
a) From your partner: the timing is right, since warts typically appear 6-12 months after exposure, about the duration of your relationship. This in turn could mean your partner had the HPV strain before she received Gardasil; or that she has a more recent infection with a type not covered by the vaccine. Although 90% of warts are caused by HPV-6 and 11, against which Gardasil is highly effective, that still leaves 10% caused by other types.
b) From some other partner, fairly recently: If you had sex with other persons in the months leading up to your current partnership, you could have been infected. Although 6-12 month onset is typical, it can take up to 24 months.
c) Some other parther, a long time ago -- i.e. perhaps your warts are recurrent, the result of an HPV infection caught any time in the distant past (since whenever you first had sex). Although late recurring warts are not the rule, it happens sometimes.
Unfortunately, you probably are never going to know which of these is the case -- and yet the answers, if they could be known, obviously would influence the answers to your specific questions. To those questions:
1) Pap smears are not a test for HPV. Many women have HPV and entirely normal pap smears. All that can be said is that your partner does not currently have abnormal cervical cells of the type that indicate HPV or a risk for future cancer. That's good to know, and should be reassuring. But it says nothing about whether or not she might currently be infected with the HPV strain causing your warts.
2) There's no point in closing the barn door after the horse has escaped. Since you have been having sex (presumably unprotected) all along, and either or both of you has probably had the HPV infection for much if not all of your sexual relationship, there is no need to alter your sexual practices now. Not only is the barn door open; the horse is galloping over distant hills.
3) Att his point, it really doesn't matter how long your visible warts persist, unless they concern you cosmetically. Some men just want them gone (the faster the better!), but in terms of transmission etc, it makes no difference. Most likely you should just continue the imiquimod -- but if in doubt, or if they seem to be responding slowly, return to the doctor to be rechecked.
4) Yes, the information about HPV and warts can be confusing. Some of this is due to physicians, website, and other sources that are just not up to speed with the science. But much of the reason is that the science itself is changing rapidly; some of yesterday's "facts" are now known to be false, and undoubtedly this will be so for some of today's apparent truths.
My main advice is to not let this be a bigger deal than it is. You and your partner are dealing with a relatively minor inconvenience, not a serious health threat to either of you. This problem also is not a commentary on your sexual fidelity to one another, or on either or your past sexual choices. Almost all sexually active people get genital HPV somewhere along the line, often several times. Sexual infection with HPV is normal -- not desirable, but pretty much inevitable and not a mark of sexual irresponsibility. Nobody's sex life ever should be dead on account of HPV, and certainly yours is not.
My guess is your partner will not develop warts, or an abnormal pap smear; that your warts will clear up; and that's the last you'll hear of it. But even if warts recur, or if she develops warts or an abnormal pap, they just need to be managed according to a qualified doctor's recommendations -- again, as an inconvenience, not a big bad deal with serious implications.
I hope this helps. Best wishes-- HHH, MD