There is no Church rule on who may or may not be present during childbirth. It depends upon the couple. As for pain relief, again, it depends upon the woman, her health, the best for the baby. Usually, no drugs is best, but this isn't always the case. Some couples like for both parents to be present, but in some cases, it happens that Dad doesn't make it in time, or, in the case of a relative of mine, Dad passed out, hit his head on a nightstand and sustained a concussion. He ended up staying overnight for observation while Mom and baby boy went home! (He has stayed in the waiting area ever since.)
The only childbirth novelty I personally think is in bad taste is to video the blessed event and show it to friends. My parents were once subjected to a video of their friends' daughter-in-law giving birth to their first grandchild. My father practically died of embarrassment, and my mother nearly passed out as she does upon the sight of anyone's blood but her own! I've been a Lamaze coach for a friend whose husband was in Iraq, and I've been present for the births of nephews and nieces, so I'm not squeamish although I am single. Nonetheless, I would not care to see a video of someone I know giving birth! I would definitely NOT want my in-laws seeing me and I'd be horrified if they showed a video to their house guests!
In the animal kingdom, the male is usually not present for the birth of his offspring, but that is often because he is prone to killing and eating them! Just my 2¢ worth.