Not legally without authority approval
If you have a passport, yes you could.
Any tourist can rent a car provided he/she has a valid passport and a means of payment (i.e.: a credit card).
To cross INTO Mexico, you need...nothing. You just walk or drive through. However, in order to get OUT of Mexico, you need a U.S Passport and it's always a good idea to carry around some form of identification, just in case.
As long as the passport is valid he or she can enter the US from whatever country they are returning from. Yes.
Your passport is the only document you need when visiting Mexico for tourism purposes.
Your client needs B: a valid passport
Yes, unless the Brazilian has a passport of the United States.
They must get their passport in the country they are in. There is no way for you to get someone else's passport. Sending them the money to get it would be the only way you can help. As far a visa is concerned (which you didn't mention) It is helpful if you write a letter inviting them so they can take the letter to the US Embassy and apply for a tourist visa in Mexico.
In the US? No, a passport is the only Mexican document recognized as ID outside of Mexico.
To fly into Mexico you HAVE to have a passport. As of right now the only way you can get into Mexico without a passport is by land travel, but that will change as of Summer 2009. As of Summer 2009 you will need a passport for any land travel into Mexico and Canada.
Most foreigners can obtain a Mexican passport if they move to Mexico with an immigrant visa and live there for four years (2 years if married to a Mexican citizen). Then one can apply for Mexican citizenship and passport.
If you are traveling from outside Mexico, all cities you travel to require a passport.