No, Mexicans are not automatically authorized to work in the US.

Also, whether they work in the US is not relevant to their tax status.  Their tax status is determined by §7701(b) and, primarily, substantial presence test, which is a simple day-counting exercise.  Their tax status, in turn, determines how their income would be taxed.  Resident aliens are taxed on worldwide basis whereas nonresident aliens are generally taxed only on US-source income.  These basic concepts are fundamental to foreign national taxation.

As I said, there are various elections and exceptions that would need to be considered.