Make the question more direct.
does a citizen of Mexico need a passport to go from the us to Mexico and back? He is legal to be in us but not a us citizen.
Since your dad is a citizen, I think you are too. You will need to prove it since you were born in Mexico.
No. Neither will automatically become citizen of the other country.
No.
yes
it would go to the legal resident of the host country
Mexico has very little citizen participation. There is so much corruption that the government and the citizens do not care for each other.
With the appropriate permits, yes.
No, if such citizen keeps his/her passport at hand.
Yes.
Of course you can. Mexico does not have movement restrictions of any kind.
no