I just recently found out that my boyfriend’s recent ex was diagnosed with HPV cervical cancer. I know that men are carriers and don’t show symptoms, how likely is it for me get infected?

I have HPV and have broken out with genital warts. I am currently 38wks pregnant and my doctor said we may have to do a C-Section because of them. How risky is it to give birth vaginally? What are the chances of my baby getting them?