I’m not sure of how influential Jane would be. She tried interfering beforehand but was put down. However, after the birth of a son she could have more sway over Henry. I do, however, think that a realignment with catholicism is out of the question. Henry was not about to bow to the pope again. The closest we could get would probably be similar to his OTL church. On paper a protestant church but in reality a catholic church with the king in charge instead of the pope, no monasteries and an English bible. However, she might influence Edward and any surviving children they might have to stick with this almost-catholic faith.
As to the gene thing, it has been theorized that Henry was rhesus positive while Anne was rhesus negative. This would explain why Elizabeth was healthy while her later pregnancies weren’t. It could, however, also simply be stress or bad luck. If one ascribes to the former theory then Jane could still easily have more healthy children, if we imagine her to be rhesus positive. If one believes that it was simply stress or bad luck then Jane could still easily have more children