As a returning resident, you can get health care in AB as soon as you arrive. However, keep in mind that in order to be eligible for health care in any province, you need to reside there for 6 months a year. If you arrive, register, have a baby and leave and do not stay six months, you take the risk that AB may retroactively cancel your health card and send you the bill for your baby. However, I have never heard about AB doing this but BC health has.
Your baby with 1 Canadian and 1 US parent will have dual citizenship no matter where it is born.
Actually, the rules for the US green card say you should not be outside the US for more than a year so if you are in Canada for less than a year, you shouldn't have to worry. You can verify this here: https://help.cbp.gov/app/answers/detail/a_id/820