I started at SWA last February (03) when my wife was 6 months pregnant with twins. They were born a month early, 3 days after IOE. My temporary domicile was BWI (near where we live) and the chiefs pulled my next trip using sick pay, giving me about 14 days off before going back to work. If you really wanted to save the sick leave and didn't need the money, they would certainly pull any trip after birth with no pay. As far as being around for the birth, my timing was lucky. Try to get trips (through trade) that stay near where you live. The chiefs said they would pull me off a trip immediately if I was traveling and labor hit.
The FMLA doesn't kick in for new hires until one year of employment, as provided for in the law, so it will not be a factor.
Bottom line - you will get whatever time you need.