Do the Mexican race contain Native American ancestry or not?

"Mexican" is an nationality just as "American" and "Canadian" are nationalities. They are not races and say nothing about ancestry or skin colour. Some Mexicans like some Americans and some Canadians have Native American ancestry but the word "Mexican" does not imply any ancestry, only that you are a citizen of Mexico.

Mexicans are not race,they can be White,Black, Mestizo, Asia,Indigenous or another background.Most Mexicans are Mestizo of mixed European and Amerindian ancestry

Yes

There is no such thing as the "Mexican race", the ancestry of the Mexican people include European, and the indigenous people of Mexico.

Mexican is not a 'race' and genealogy is about researching individual ancestors via written records to prove what ancestry that individual person has or not, it is not about a whole nationality of people

That would depend on how you define the term "Mexican race" as citizens of Mexico come in all the same combinations as Americans do, just usually in different proportions.