What you describe is pretty typical for the renix power distribution system.
I think 8mud is probably right here, you're in ohio, it's f'ing cold there, lead acid batteries work very poorly once that temp drops below <40. So if you have an older battery this could very easily be your culprit.
However, before you go out and buy a new battery, check your cable connections, take the battery clamps off and clean the wire ends with a bronze brush and some contact cleaner, do the same thing with the clamps. Besides cold temps, dry air has a tendency to cause your contacts to go bad, this creates resistance that will ruin your day just as much as a dead battery.
The thing to remember is that the starter has a direct line between the battery and the lead on the motor, if the starter isn't cranking the motor over, then the problem exists somewhere in there.
Basics of renix debugging:
1) If the engine doesn't crank, it's either battery or wiring
2) if the engine cranks, but doesn't start, check for spark, if no spark it's your CPS or CPS wiring (usually it's the wiring)
3) If the starts, but idle goes high, your IAC/IAT system is dirty, if not that, your TPS is not working properly.
3a) If it starts, and then dies when you start driving it, your IAC/IAT system is dirty
Also, it is not your coolant temp sensor, the only way this could cause problems is the engine temp affects the timing advance controlled by the ECU. When a car is started (hot or cold) it's running in open-loop mode, until the ecu puts it into closed-loop. What would happen if it was the temp sensor is your car would never go into closed loop (the ECU would think it's too cold).
Given that you swapped the battery, I think this means either you contacts are dirty, or your battery is hosed.