Insurance and registration back in the old country: Your car in México must have the license plates that were issued with your registration that you had when you arrived in México.  (Or new plates if you change your registration.)  From that time on, México has no interest in whether you keep that out-of-country registration up to date.  Likewise they don't care what you do with your non-Mexican insurance.  They are only interested in your plates for identification. Your "registration" in México is the permit you were issued at the border.

Unless you need to drive back to the old country from time to time, you can probably save a goodly sum by letting registration lapse and canceling the insurance. 

If you return to the old country to live, all insurance companies will charge you a penalty for being uninsured for some period of time.  Before you leave, talk with your insurance agent.  He may be able to offer you a non-operating or storage policy that will keep you on the rolls for only a few dollars per year.  Some of these holding policies also allow for short term re-activations when you want to return north for a visit.

There are some states (NY, FL, perhaps others) that will suspend your driver's license if you do not maintain insurance on your car.  This can cause you great problems if you want to drive in the USA at any time.  (Unless you get a Mexican driver's license; it is recognized in the USA just as your USA license is OK in México.)  In those states you need to arrange non-operating registration and insurance to save you driver's license. Or you need to register your car in another state with easier rules.

A few years ago, if one wanted to drive back NoB with expired plates, it was possible to get a temporary permit to cover your expired license plates and to buy trip insurance.  Those days are gone.  It is no longer possible to work around expired plates or to get trip insurance.  So if you are living in México with expired plates, forget about ever driving north again.  If your plates are still good, and if you can reactivated your insurance, you can still make the trip

Personally, you couldn't pay me to drive back.  Fly and rent a car, it's cheaper and a lot less painful.

Some Rules for Your Car:

Who can drive my car?  Anyone with a valid driver's license can drive your car IF YOU ARE IN THE CAR.  If you are not present, the car can only be driven by family members and other foreigners..  Read Article 106.  No matter how pressing the circumstances, don't loan your car to your Mexican friend, or gardener, or maid. etc.  What is the penalty for breaking this rule? -- the confiscation of your car. 

Can I use my car to make money?  No.  Any for-hire use of the car is a violation of the permit.  Consequence of cheating? -- confiscation of your car.

Do I need a Mexican driver's license?  No, if you have a valid license from the old country, it is OK in México.  However, as the lady in my local license office explained to me, while the law says you don't need a Mexican license, some cops don't understand that and having a Mexican license may save a hassle.  I have one.  It was very easy to get.  Since my California license was still valid, I did not have to take a written or driving test.  Nor a vision test.  All I had to do was fill out an application, have my picture taken in the office, pay some money, and, strangely, go to a local clinic to have a blood type test.  My blood type appears on my license.  Actually I think that is a good idea.  Driver's licensing is a state matter, so the rules vary about what tests you may have to take and how much it costs. 

The states of Morelos, Puebla and Guerrero require only a tourist visa (Visitante). The states of Durango and Coahuila will issue a license only to a Residente Permanente. All the other states require an Residente Temporal or Permanente

Can I own a Mexican plated car?  Yes.  
In addition to my foreign plated car?  Yes.

What if my car is stolen?  If it is recovered, you only need to deal with the local police and your insurance company.  If it is not recovered, you will also need to deal with the Aduana/Banjercito.  The basic view of the government is that an un-recovered stolen car could also be a car that you sold and then reported stolen.  Unless they have a real reason to suspect you of the crime, they won't make a point of it, but you will have to pay a "tax" on the car that you cannot account for.  Often the tax can be settled for 40% of what you get from your insurance company.  You will have to pay this tax before you can clear the stolen car from the computer and thus be allowed to bring another car.  If bringing in another is not an issue, then I guess you could just let the car ride on the records.  I hope I don't ever have any first-hand experience with this.

What if my car is destroyed in a wreck?  There is a procedure for removing it from the records.  Go to you local Aduana office for the forms and instructions.

What if my car is broken down and not worth repairing?  There is a procedure where by you can give the car to the Mexican government, and that gets it off the records..  Go see the Aduana.

The Mexican government is very gung-ho about their car rules.  Don't mess with them; you'll loose.

Some Rules of the Road:

If you hit a bicycle, it's your fault no matter what.

All states allow a right turn on red.  But not all cities allow it, so inquire locally.

Seat belts are required by most states, but not all, so ask. Or use them to live longer.

You will find that street signs and stop signs are not in great supply in Mexican towns and cities.  This makes using a map a real chore.  It also requires that you understand the local customs in regard to "implied" stop signs.  For example, in Lerdo (my city) all north/south streets have the right-of-way, and all traffic on east/west streets must stop at every intersection whether there is a stop sign or not.  There are a few streets that are exceptions to this rule.  They are well marked with stop signs, thank goodness.  In many larger cities the rules for implied stops are different in different parts of town.  When in doubt, give the other guy the right-of-way -- you'll live longer.

Where there are street signs, they are often on the side of a building rather than on a pole by the side of the street.  The sign is typically a single line with the name of the street, and an arrow indicating the direction of traffic.  The color of the sign acts as a traffic sign -- red=stop, green=right-of-way.