You are not supposed to turn off the TCS when driving on snow covered roads, the only time TCS should be turned off is if you are stuck in snow, or mud.
How worn are your tires?
Meguiar's has a Clear Coat Safe Rubbing Compound so that would be a good choice as it is supposed to be much less abrasive than the older rubbing compounds.
https://www.meguiars.com/automotive/products/clear-coat-safe-rubbing-compound
Could just be a faulty battery that won't hold a charge. You can take it to Autozone to load test it for you to find out if is faulty. Sometimes a cell will go bad in them even in newer batteries.
Possibly a dirty or faulty fuel injector on cylinder 3. Try some fuel injector cleaner first before replacing the fuel injector.
You can try swapping the fuel injector on cylinder 3 to a different cylinder, first clear the CEL with a code reader and if the code is now on the cylinder you...
Actually it is a very common occurrence on not just Mazda vehicles but many other manufacturers vehicles.
Do a google search and you will see what I am talking about.