Here is a wiring diagram Note it is for a GW7, I could not find one for a GW5
http://manuals.frigidaire.com/prodinfo_pdf/StCloud/2165839.pdf

[COLOR="Blue"]Is it possible that the defrost cycle is interfering with my hardwired Johnson Controls Thermostat causing these bizarre compressor issues?[/COLOR]
Possibly but I do not know how you wired your new thermostat into the unit.

[COLOR="Blue"]Could somebody please teach me how what modes I am switching between with the defrost timer knob?[/COLOR]
It is either defrost heater on or compressor and fan on.
I am assuming from looking at the parts breakdown that this is an evaporator fan used to circulate the air in the unit over the evaporator coils.

[COLOR="Blue"]Should the freezer always be on compressor mode for thermostat control to function properly?[/COLOR]
Not sure what you mean here.
Looks like the thermostat not only controls the compressor but also the defrost timer motor so if the thermostat is not closed the timer will never advance out of the defrost cycle and move to the compressor run side of the defrost relay.

[COLOR="Blue"]If so, is there any way I can bypass the timer so that the freezer doesn't start a defrost cycle?[/COLOR]
Yes it is possible just place the timer cam in the center of a cool cycle and disconnect the timer's motor. Problem is you will have to manually defrost the unit.

[COLOR="Blue"]I am making a lot of guesses and assumptions on not a whole lot of knowledge, so don't hold back if you think I might be barking up the wrong tree with this theory.[/COLOR]
The problem is that the food in the freezer acts like a large cold reservoir so this stabilizes the temperature. With that removed you are likely to have fast temperature swings as it tries to regulate the temperature. Also you compressor is on full or totally off so that will also contribute to fast cycling. It also depends on the temperature variation of your new thermostat. For instance it it opens at 35 and closes at 35.5 it will cycle quickly forever.

Also there is a problem with the temperature gradient in the unit since the fan only circulates when the compressor is on you could find it much warmer at the top of the unit as the cold air will drop to the bottom.

Looks like you may have some experimenting ahead of you.

I have never done this so am just throwing out possible problems/solutions.
I sat down and gave this another think.

I looked at your unit parts and this is what I would do if I was tackling this.
I would remove the cover from the evaporator coil. It should then not hold the cold air in this area and therefore you should not need to defrost it as it should melt off any frost due to your thermostat's set temperature.

I would then by pass the timer and internal thermostat by connecting the yellow wire from the compressor and fan (see wiring diagram directly too your power cord (black).
Be sure that you new thermostat can handle at least 15 amps.

You may have to jiggery pokery a mount for the evaporator fan so it pulls air through the evaporator coils.

[COLOR="Blue"]Currently the freezer is working. What seems strange to me is how unreliable this freezer is. It will work beautifully for a few hours and then the compressor will seize up and exhibit symptoms of a burnt relay. The temperature shoots up and the compressor will click on (dimming the lights) run for 5 seconds and than halt with another click. Normal compressor action does not dim the lights in my apartment. Oddly, the way I fix this problem is to give the defrost cycle a very imprecise spin, and for some reason the compressor starts to behave normally again.[/COLOR]
Could be the unit is on it's last legs but your fix of spinning the defrost cam may also indicate that the defrost timer contacts are pitted and not making good contact all the time. This would then drop the voltage to the compressor and it would draw more current trying to start up.
[COLOR="Blue"]
I am confused by your cold reservoir comment. Are you saying that it will be difficult to use a freezer as a refrigerator because a freezer compressor is usually has the help frozen foods to stabilize the temperature?[/COLOR]
Yes. On most fridges you measure the temp in the fresh food but is is fed by the cold air from the freezer so the food in the freezer acts as a cold supply so it does not turn on the compressor as often. Hope this makes sense!

[COLOR="Blue"] I do have a large metal keg inside the freezer... will that help maintain the temperature?[/COLOR]
Only time will tell if this will work but your keg should act as a heat sink to stabilize the temperature the same as the food in the freezer

[COLOR="Blue"]Have you ever heard of anybody fixing compressor halting by advancing the defrost timer?[/COLOR]
No but see above. It also could be the relay.

Another possibly important point is where you locate the thermostat's sensor.
I would start in the vertical center of the unit first.
Would be a good idea to get a thermometer so you can check the temperatures at the top and bottom of the unit.
You do not want it freezing on the bottom of the keg. If the keg is near full the liquid should circulate by itself to prevent freezing but if the keg is near empty it could freeze.

One other thought I had is you may want to get a small pancake fan and put it in the freezer letting it run all the time so it circulates the air and prevents temperature gradients. You would not need much air flow so should be efficient as far as electrical use.

PS: I love doing things like this. At the lake my fridge is a fridge in summer and a hot box in winter so I do not have to hump everything back (that may freeze and explode) to the city.

Hope this helps, good luck and have fun.

[COLOR="Blue"]Do you think that ordering new defrost timer might help?[/COLOR]
It could but I cannot be 100% sure, just seems to me that if playing with the timer fixes the problem then it is the most likely culprit.
The only way to tell for sure is to measure across the defrost timer contacts when it is happening. If you see a voltage there then you know that the contacts are dropping the voltage to the compressor and causing the problem.

Note if you are going to leave the defrost timer in circuit do not remove the evaporator cover as will cause too much heat to go into the main compartment.