Car won't start. Battery not taking jump?

sounds like you have a failing ecu. the capacitors in the ecu are worn out. there has been a lot of problems with the ecu on mazda's over the years, only take one out put overload from the alternator and the ecu starts playing up. no codes come up as it don't code its self. having the battery off is like a reset.

may be a poor contact.
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MD802

Take it in to have a charge system check done on it, Your alternator is probally shot,

My cruise control was not working for some time. I replaced the brake pedal switch because it was cheap but to no avail. Tried a Master Reset in the System Settings and that did nothing. Tried a hard reset by disconnecting low voltage battery positive terminals. That did nothing. Then, one morning I came out and the car was totally dead. Using the mechanical key, I got in and then jump started the car. It turns out that someone in the family had changed the headlight setting from Automatic to On. That caused the battery to run down overnight. Long story made short is that after jump starting the car and driving it for a while to get the battery charged, the cruise control worked perfectly and has continued to do so for several weeks. I do not know why this accidental happening fixed the cruise control so I share this in amazement.

Need to get a meter on that car,auto zone will do it,make sure it's charging.
I am thinking the voltage regulator is not in the alternator.Think it's in the control computer. If it goes bad ,the car won't charge.Instead of buying a computer you can also use an older style (say 1980's) external electronic regulator.have to get a pigtail from a parts store ,hook up 2 wires to the alt. Cap and seal the old wires that used to go to the ECM. Works fine and won't go bad again.

The body control module is the problem it is drawing the battery voltage down,and causing the lighting malfunction

Battery is dead ... dropped a cell ... get a new one ... the cold weather is hard on poor old batteries

replace the alternator,this is the problem,please rate thank you.

remove and, clean your battery cable clamps then retighten and, test.
If your battery is too discharged "jump" it using jumper cables and, another car.

Check for a parasitic draw?