What are the steps to bringing my cousin from Mexico to the US as a tourist? And can he then apply to stay here?

There is no true way for you to petition for him. Instead he applies himself at the US consulate in Mexico. They will analyze whether or not they think a)he has a legitimate reason for coming to the US and b) whether he will return to Mexico at the end of his trip. Having an invitation from you can be good evidence as to reason for stay, but he will need to gather very strong evidence that he will return to Mexico as well.

You cannot petition for a cousin as it is not a family relationship that qualifies for a family-based visa. He/s he can apply for a tourist visa if he/she is able to show that he/she has no immigrant intent and only wants to come visit the US for a brief period of time.

You may write him a letter inviting him to come visit you in US for a temporary period. But, he himself must go the US embassy in Mexico and apply for a tourist visa. The consulate is often very dubious of young men applying to come to the US on for visit. There is a presumption that such young men are economic migrants looking for work in the US. Once in this country, such men will likely not return to their country. In order for him to convince the Consulate otherwise, he must show existence of strong ties in his country that will ensure his return. Such evidence include ownership of property in his country, family ties, his obligation to take care of aged relatives, a secure job that he cannot afford to give up, If he is employed, his desire to visit US must coincide with a planned vacation authorized by his employer (preferably in writing) He must also demonstrate that he has sufficient money for the trip, as well as a round trip ticket Your letter to him should include your relationship to him, that you will be responsible for his food and lodging and other incidentals while he is visiting you in the US and your personal guarantee that he will return to Mexico.

It really does not depend so much on you, but on his being able to overcome the presumption that if he comes in as a tourist, he will remain in the US. That is a universal presumption that must be overcome by all tourists applying from non-visa waiver countries. It is your cousin that needs to show that he has a steady job or goes to college, perhaps that he has a family, owns a house, owns a business, etc.