The answer is yes, to both questions. To the second, of course you can leave your car in the US when going home for visits. It just needs to have met all the legal and regulatory requirements (tags, inspections, emissions, gas consumption, and any applicable import fees, etc.).
To the much larger question, bringing your personal vehicle from Mexico to the United States, for any time period longer than those allowed as a visitor, could be considered importing the vehicle.
You would have to do a fair bit of homework in advance, and the USCIS has detailed information, with this an overview:
Importing a Motor Vehicle
The Imported motor vehicles are subject to safety standards under the Motor Vehicle Safety Act of 1966, revised under the Imported Vehicle Safety Compliance Act of 1988; to bumper standards under the Motor Vehicle Information and Cost Savings Act of 1972, which became effective in 1978; and to air pollution control standards under the Clean Air Act of 1968, as amended in 1977, and 1990.
If vehicles manufactured abroad conform to U.S. safety, bumper, and emission standards, it is because these vehicles are exported for sale in the United States. Therefore, it is unlikely that a vehicle obtained abroad meets all relevant standards. Be skeptical of claims by a foreign dealer or other seller that a vehicle meets these standards or can readily be brought into compliance. Vehicles entering the United States that do not conform with U.S. safety standards must be brought into compliance, exported, or destroyed.
This pamphlet provides essential information for U.S. residents, military or civilian government employees, and foreign nationals who are importing a vehicle into the U.S. It includes U.S. Customs and Border Protection (CBP) requirements and those of other agencies whose regulations we enforce. Since Environmental Protection Agency (EPA) and Department of Transportation (DOT) requirements are subject to change, we recommend that you contact these agencies before buying a vehicle abroad.
EPA has a detailed automotive fact manual describing emission requirements for imported vehicles. You may obtain a copy of this manual, called the Automotive Imports Facts Manual, or other information about importing motor vehicles by calling EPA's Imports Hotline at (734) 214-4100. EPA's page on Importing Vehicles and Engines contains additional information.
You may reach DOT's vehicle hotline at (202) 366-5291 or communicate by fax at (202) 366-1024. The DOT website can provide further assistance.
Nonresidents may import a vehicle duty-free for personal use up to (1) one year if the vehicle is imported in conjunction with the owner's arrival. Vehicles imported under this provision that do not conform to U.S. safety and emission standards must be exported within one year and may not be sold in the U.S. There is no exemption or extension of the export requirements.
Driver's Plates and Permits
Imported cars should bear the International Registration Marker. The International Driving Permit, issued in five languages, is a valuable asset. Consult an international automobile federation or your local automobile club about these documents.
U.S. residents importing a new or used car should consult the Department of Motor Vehicles (DMV) in their state of residence about temporary license plates and what documentation their DMV would require from CBP.
Nationals of Central and South American countries that have ratified the Inter-American Convention of 1943 may drive their cars in the U.S. for touring purposes for one year or for the period of the validity of the documents, whichever is shorter, without U.S. license plates or U.S. driver's permits, provided the car carries the International Registration Marker and registration card, and the driver has the International Driving Permit.
Motorists visiting the United States as tourists from countries that have ratified the Convention on International Road Traffic of 1949 may drive in the U.S. for one year with their own national license plates (registration tags) on their own national license plates (registration tags) on their cars and with their own personal drivers' licenses.
Motorists from Canada and Mexico are permitted to tour in the U.S. without U.S. license plates or U.S. driver's permits, under agreements between the United States and these countries.
Motorists from a country not a party to any of the above agreements must secure a driving permit in the U.S. after taking an examination.
Foreign nationals employed in the U.S. may use their foreign license tags from the port of entry to their destination in the U.S.