Fix the electrical issues first.

Okay, so I cleaned off as much of the white corrosion as I could, and a similar thing happened to when I changed the air filter. It ran for a few minutes and died.

Check for codes

Alright, so I've checked the codes again and I've got
p0113
p0750
p0753
p0758
p1655
p1675

The vehicle seems to have gotten more predictable after cleaning the battery.

It will die after about ten to fifteen minutes of idling and then won't start again for another thirty minutes or so, which it will do very quickly.
This pattern has repeated itself three times so far.

You have an electrical issue somewhere. Seems like something has eaten your harness or you have ground issues somewhere.

Okay, so that would help explain why it happened when it got cold, I suppose.

So I've purchased a spark plug tester to test the ignition system. What other systems are there?

I really need a little more direction. Other than hoping it'* a bad spark plug cable, I'm not sure what else to look for.

A bad spark plug cable will not cause all of those codes. You need to visually inspect the PCM harness to each of those components listed in those P codes you posted.

Okay just the harness leading to the two shift solenoids, the Evap purge solenoid, the TCC solenoid and the Evap Vent solenoid?

Alright, thanks, this is a lot of stuff to track. Really can't do anything about it til Sunday due to work.

So where can I find where all this stuff is located and where all the wiring to it?
How obvious is what I'm looking for?

How obvious is what

I would take the battery in to your local parts store and have them load test it. There will be no charge for this. The symptoms you are describing and all the codes it is throwing is telling me something fundamental with the electric system has failed. Mike is right in that it might be a wiring issue, but since checking the battery is free you should start there.