There are no US based carriers in Mexico.
Each US carrier has roaming agreements with Mexican carriers. AT&T is usually a ripoff and the most expensive option. T-Mobile is usually the cheapest but can impose limits on data speeds. They all use the same towers on the island so you’re really looking at price and speed differentials. Do the research on the carrier website. Buying a local SIM is always the cheapest but most inconvenient option.
Not true.
We actually have an AT & T store here and AT & T is an option here as a local carrier (for 3 - 4 years at least) although they have to compete with Telcel. I've had my Telcel line for 18 years and have a great plan therefore no reason to switch so I don't really know what plans they have.
I also have a T-Mobile line for my US number. Depending on where I am on the island, I'll either get a Telcel or an AT & T signal.