Are military members required by regulation to care for their unborn children?

Morally obligated, but we know how that goes. Once the child is born, the mother can go through the court order process. But in 3rd world countries the odds of that are extremely small to non-existent. The more industrialized the country, the better chance of the mother being able to get "non-voluntary" support.
Unfortunately many SM's "hit it and quit it" while overseas and don't give a crap about the child they leave behind. It's not their "problem". On the other hand, there are some local girls who will try to get pregnant by an American SM in an effort to get to "the land of milk and honey".

No. Because, until the child is proven to be his thru DNA testing, he is not obligated to do anything.

Nothing in the Regs I'm aware of. However, the MIL side will comply with any valid court order for child expense, etc. including pay garnishment. She'd have to go through the paternity mumbo jumbo which may be too onerous if you don't have deep enough pockets.