How easily can a new baby be infected with HPV after birth, if both parents are infected?

Neonatal hpv. If the mother is infected and active lesions of hpv in the birth canal, the fetus during the birthing is exposed to the hpv virus and can be 'inoculated' with the virus by the rubbing action brought forth by the vaginal delivery. The baby can get vocal cord hpv or even genital hpv. However this is a very rare event. So if a small baby has genital warts one has to also rule out sexual abuse.
Rare if at all. The only marker we have for newborn acquisition of HPV is genital or vocal cord warts that come up within 2 years of birth. These might be expected if mom has active warts around the time of delivery. We have no markers or experience with non-wart genital strains.
Not easily. 85% of adults have hpv (whether they know it or not). That being said, it is extremely rare for a baby to contract hpv at delivery. There are a few cases reported in the literature of hpv being found on the vocal cords of children, but as I said, very rarely. And, if 85% of us have hpv, then almost every baby is exposed to hpv at delivery and you rarely ever hear of a case being transmitted.