For example, abortions may not be allowed in my state, but if I go to a state that does allow them, could they perform it on me or would they see what state I'm from and then deny me. Another example, the age to drink here is 21 but if I go to Mexico and the age is supposedly lower, would it be legal for me to drink there then even though I'm from a state with a law that says 21?

I don't think they would be checking to see what laws your resident state has regarding the issue of abortion. As for drinking in Mexico, you would follow their laws; they don't care actually, how old you are.