Well, this depends on what you mean by 'truly protected'. Virtually every safety device that exists is not 100% protective: people die in auto accidents while wearing seat belts, batteries fail in smoke alarms, etc, etc. That doesn't mean we should stop recommending seat belts and smoke alarms.
Condoms provide a high degree of protection against HPV and HSV. But the protection is less than for HIV, gonorrhea or chlamydia; those infections require exchange of secretions, but HPV and HSV are transmitted skin-to-skin, so there is opporunity for contact above the condom range.
1) The data on herpes are not sufficiently precise to answer your question. Of course the risk depends on the infection status of one's partner, and even that isn't straightforward: symptomatic infection? how long since that person was infected (the shorter the time since infection, the higher the risk of transmission)? asymptomatic shedding? But even without condoms, in couples in which one person has HSV-2 who have sex 2-3 times a week, WITHOUT condoms, the infection is transmitted in 3-5% of couples per year. So the risk per exposure is obviously low. As a generalization, that risk probably is reduced by 80-90% when a condom is used.
2) The cellular proliferation that causes pap abnormalities and cancer occur primarily in the cervix, but the infection involves any and all genital surfaces--penis, vulva, etc. And yes, you could well have a high risk infection that could be transmitted to a partner despite condom use.
Condoms are less effective against HPV than herpes. The best research shows 70% reduction in risk of aquiring HPV in female college students who use condoms 100% of the time. Those who use condoms 55-99% of the time had a 50% reduction in HPV risk. At less than 55% use, there was no protection. The bottom line is that 100% consistent condom use will help protect against HPV, and it reinforces the advice that all sexually active people outside permanent, monogamous relationships should use condoms. But in the long run, most people can still expect to catch HPV, mostly with the high risk types. You should look on this as a normal event, not much different than carrying staph or strep on our skin.
As an epidemiologist, you undoubtedly know how to search the medical literature. Look for recent papers by Anna Wald and her colleagues on HSV and condoms, and Rachel Winer and colleages for the masterful research study on HPV and condoms.
Regards-- HHH, MD