Hello, can you be certain that the engine itself turns, manually, with a breaker bar and socket on the crankshaft? If the engine had been "running hot out of the blue" what if there was a catastrophic failure internally causing the engine to seize?
Being that I don't know the vehicle first hand, we have to start with things like that, and go from there.
Besides ensuring that the battery is fully charged and able to accept a load, you will need to check the battery cables to make sure they are free from corrosion and secured well. You will also need to check any of the fuses to make sure that none are blown. These could be high amperage maxi-fuses in the box under the hood.
Beyond that, if it is an automatic transmission you will want to make sure that the shift lever is securely in park or neutral gear. This can be measured using a Mazda compatible high end diagnostic tool.
If you have a manual transmission, the clutch switch should be checked to make sure it works.
Lastly, the anti-theft key system should be checked to make sure the key you are using is properly programed for the vehicle.
Use this link to find some shops near you that can help with your concern:
www.repairpal.com/repair-shops
