What is the relationship between the Catholic churches of various countries? I currently live in Mexico, where the Catholic church is quite different than the Catholic church I'm accustomed to in the U.S.  This is most obvious to me by the hostility often shown by Catholics (especially clergy) against non-Catholic Christians.  As an anecdotal example, some good friends of mine were actually ex-communicated from the Catholic church in Mexico for refusing to end a Bible study they held in their home with non-Catholic Christians. Stories like this in turn mean that many non-Catholic Christians in Mexico have a very negative view of the Catholic church here, as well.

As I've tried to explore and understand the various differences between the U.S. Catholic church and the Mexican Catholic church, I'm left not really knowing where to start asking questions.

So my first question here is:  How are the Catholic churches in various countries related, and how do they differ?
Countries or regions each have their own bishops conference. 

Conferencia del Episcopado Mexicano

United States Conference of Catholic Bishops

But that is as much as they differ.  A bishop, acting as a successor of the apostles (like James as Bishop of Jerusalem) is the leader of his diocese.  Then the priest is the leader of his congregation and establishes the norms for conduct within his parish (i.e. if he doesn't want a bible study to be held there, then there shouldn't be a bible study held there).

To your anecdote, I'm not a canon law expert, but it strikes me as odd if it were a priest telling someone that they're excommunicated.  He may tell them to take a hike, but I don't believe that he has authority to excommunicate someone.  There are offenses, like abortion or helping someone get/perform an abortion that are in themselves automatic excommunication, but what you're describing sounds like disobedience which would require a Bishop's intervention.

In my diocese, we had a similar thing happen with a Religious Ed. teacher who refused to deny things she had written in her master's thesis which were contrary to Catholic Teaching.  To my knowledge, she wasn't even excommunicated for that, just fired (you can read more info than you'd ever want here, but I think it should clarify a few things about the way a diocese is governed). 

What Gilbert wrote is perhaps the way things look, but in truth, it is not the congregation who leads things in their parish.  Throughout the Universal Church, priests are the heads of their parishes.  

In the documents of Vatican II, the Church provided for additional lay leadership from within the parish in establishing ministries and financial councils but not complete authority and those documents cover the entire Church, not just Mexico or the United States.