In California, when there is no court order regarding custody in place, both parents have equal rights to the children. However, California law requires written consent from the other parent before removing minor children from the state, whether for relocation or vacations.
Mexico requires (but does not always enforce) that you have a notarized letter in English and Spanish from the other parent allowing the child to travel with one parent.
No. One needs a valid passport to travel internationally. A birth certificate is no substitute and ads nothing.
It's important that you give your co-parent notice and preferably get their express permission in writing if you wish to travel outside the state with your child. If the other parent refuses to cooperate, you can ask for the court's permission in advance.
You should obtain a child travel consent letter granting you permission from the absent parent to leave the country with your child (having it notarized is not required, but highly recommended).