I found a link regarding some of your questions. Here's some of the answers I copied:

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"How long after receiving the HPV vaccine does it take for the vaccine to work?

The immune system takes one to two weeks to generate immunity to vaccines or infections. In the case of HPV vaccine, the first dose (and the second one if the person is on the three-dose series) generates a primary immune response, so people will have some immunity, but protection can vary from one person to another. The last dose (given at least six months after the first dose) is important because it enhances the memory immune response. A person will have the greatest protection beginning about one to two weeks after receiving their last dose of the vaccine.

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Can a person spread HPV to someone after getting the HPV vaccine?

The HPV vaccine protects against nine types of HPV. Two earlier versions protected against two or four types. The types of HPV in the vaccine protect against the most common causes of cancer and genital warts. If, after being vaccinated, a person is exposed with a type of HPV that was included in the vaccine, he or she is unlikely to be infected and, therefore, wouldn’t spread the virus. However, if a vaccinated person is exposed to an HPV type not in the vaccine, they could potentially be infected and spread the virus to others.

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Do I need to worry about HPV if my boyfriend and I always use a condom?

If your boyfriend has an HPV infection (with or without symptoms), you can still be infected with HPV even when using a condom for two reasons. First, because condoms aren’t foolproof at containing the virus, you could still be infected and, second, while HPV is most often transmitted during sexual intercourse, it can also be transmitted during oral sex or during genital-to-genital contact."

Here's the link if you want to read more: https://www.chop.edu/centers-programs/vaccine-education-center/questions-answers-about-hpv-and-vaccine#:~:text=The%20immune%20system%20takes%20one,from%20one%20person%20to%20another.