See the attachment for the wiring diagram.

I think you shorted out the control thermostat which would keep the compressor on all the time.

I do not think you made the defrost cycle worse but you will have to track down what is making the noise.

Hopefully you can replace the timer ASAP.

I am assuming that the only problem now is that the defrost heater did not come on during your test.
And that the compressor is running and the unit is cooling down.
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1) If the bimetal thermostat was always open and the heater thus never came on, but the defrost timer was functional, what kind of symptoms would this have?[/COLOR]
The unit would never defrost so the coils would clog solid with ice/frost.
The evaporator fan could not pull air through the coils so the temperatures would rise. The compressor then would run all the time except when the unit goes through a defrost cycle.

[COLOR="Blue"]Wouldn't it just never defrost but still keep the fridge cold and running?[/COLOR]
Yes
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2) Was I wrong to not wait until the fridge got cold enough?[/COLOR]
It is easier to check the defrost when the unit is cold/frozen.
I would have let it run to see if it will cool down to the correct temperatures.

[COLOR="Blue"]If the fridge was warm, and I set it to defrost mode via the timer, would the heater element turn on? [/COLOR]
No


3) Is there a case where the bimetal thermostat being always open would result in the fridge not running?
No

[COLOR="Blue"]4) I feel very strongly that this is caused by something in the defrost controls, but could I be missing something else?[/COLOR]
I am not sure what the problem is.
You said the fridge did run when you first installed the timer so I am confused.

[COLOR="Blue"]5) I feel like I've ruled out that the compressor is overheating and shutting off because it worked enough to freeze everything when i jumpered pins 1 and 3.[/COLOR]
I agree with you

[COLOR="Blue"]Anyways, sorry for the long post but let me know if you can figure out what's going on.[/COLOR]
Not a problem.
Slowly move the timer till the compressor starts up again and leave it to see what happens.

PS: I am not a fan of immersing a defrost thermostat in water. If any water gets into the part it will be screwed. I either freeze it in another freezer or hold it's metal face onto an ice cube.

You are welcome.

Sounds like you have got it beat but I will keep my fingers crossed as I do not want to jinx you.

[COLOR="Blue"]I put in a new defrost timer, a new bimetal thermostat, and I verified that the heater could turn on.[/COLOR]
I do not think that your problem is in the defrost circuitry.
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The only other things I can think of are the thermostat, the start relay, or the control board. [/COLOR]
The unit does not have a control board so that leaves the start device and the thermostat.
Out of those two the thermostat is the more likely.
Have you tried turning the thermostat up/down without turning it to off?
If it starts up then I would replace the thermostat.
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This is really puzzling me because it worked for a week without any problems.[/COLOR]
It is not uncommon for a thermostat to stick open intermittently.

[COLOR="Blue"]Does this add to your suspicions of a faulty thermostat/cold control?[/COLOR]
Not really

[COLOR="Blue"]Is there a difference between problems with the start relay vs the thermostat, or a way to test it?[/COLOR]
Is the evaporator fan on but the compressor is off?
The reason this is important is that they are wired in parallel so both should be on together.

So if the fan is on and the compressor is not it points towards the start device.

If the fan and compressor are both off it points towards a bad thermostat.

You can double check the thermostat by unplugging the unit and shorting the two thermostat contact wires together. You may have to tape them so they do not short to anything else. Then plug the unit in.
If the compressor fires up then it is a thermostat problem.
The unplug and plug in a couple times just to be sure.