Yes. If you have genital warts there are some that would deliver by c-section to avoid spread to the baby at delivery. That is a bit drastic but done in some locals. Babies exposed to genital warts at birth have a risk of acquiring vocal cord warts and or genital warts, which may not be evident for up to three years. Genital warts appearing after that time bring on questions of sexual exploitation.
Yes. May require treatment but yes you can have children.
Yes. Yes. They are typically treated if the effect birth.