If the Mexican Embassy has to be notified when one of its nationals is arrested in the USA, then that is prima facie evidence that the person is not fully “subject to the jurisdiction thereof”.
No. The right to consult with the Consulate comes from the Vienna Convention on Consular Relations, a multinational treaty to which the U.S. is a party.
The foreign Consulate has no jurisdiction over the arrested national; the foreign national has a right to consult the Consulate just as he or she has the right to an attorney.
The foreign national is fully subject to the jurisdiction of the U.S. unless they have diplomatic immunity of some form.
Texas just executed a Mexican national. I'd say that's evidence that they are.