There's actually a lot of research on the languages families of North America.

It also should be noted that many of the indigenous languages of the continent are still spoken by millions of people and still continue to evolve.

To answer your first question, No, the Aztec language (Nahuatl) is not related to any of the Mayan languages. The Mayan languages aren't even a single language, but rather their own unique family that aren't related to any other. There are currently about 30 extant Mayan languages last time I checked. Most Mayan languages aren't even mutually intelligible with one another.

You might actually be surprised which languages are actually related to the Aztec language of Nahuatl. Nahuatl is a member of the Uto-Aztecan language family, which is quite possibly the most studied and documented language family in North America. It also spans an incredible range, with members of the family like Ute as far north as Utah (as the name would imply) and the Pipil language of El salvador, which is a close relative of Nahuatl. This family also includes the Hopi language, and many other of the indigenous languages of northern Mexico and the southwestern United States.

Other well known languages in Mexico such as Mixtec and Zapotec are distantly related members of the Oto-Manguean language family, however, they occupy their own branches on the tree and are not mutually intelligible. There are of course other branches and languages, but Mexico and North America in general is such a linguistically diverse place that it would be very difficult to document the various languages families and isolates that exist in a single Reddit post.

It should be noted however that although the languages of mesoamerica aren't always members of the same branch or even related at all, they form what is known as a sprachbund, which means that although the languages might not be related or have the same phonemes, they may often use similar phonological conventions simply due to their proximity to one another. This is comparable to the way the languages of the Indian subcontinent sometimes sound similar even though many of the langues have No relation to one another.

When it comes to South America it's completely different however. Not only because indigenous languages are more widely spoken there, but also because they don't have any relation to the ones spoken north America and I sadly not quite as well versed in the languages of South America as I am with the ones in North America. Hopefully this post answers your question though.