Hi Chad,
Mebbe I can be of assistance?
You said: "If my component uses a standard grounded power cord, I would imagine grounding is a simple issue - everything that gets grounded goes to the third pin."
Firstly, lets first make a bedrock statement:
* a "safety earth" is often important. This means the "earth" wire in a 3-wire power cord is connected to the chassis. It's called a "safety earth" because this chassis mains-earth connection will save your life if, by some mishap, the "live" power wire ever touched the chassis.
I personally believe the only reason hi-fi mfrs use a 2-wire power cord and not a 3-wire power cord is to save themselves 10c - ie. I believe all components should have an earth wire coming in. Certainly, some other people disagree with me!
However, you said:
a) you are using a plastic case, and
b) the Sonic T-amp is battery powered.
So "safety earthing" is not relevant! However, you may find when you've got it going that your amp sounds a bit harsh ... by using a plastic - not metal - case, you've lost the protection from RFI which a metal case ("Faraday cage") gives.
Secondly, in my universe (and my whole system: head amp, phono stage, pre amp, active crossovers, power amps is DIY) no part of the sound chain is connected to (mains-earthed) chassis. IE. the "ground" on each PCB inside each component all "float" wrt the chassis. AFAIAC, this way you minimise "ground loops" - ie. hum - yet get maximum protection against RFI.
You said: "Currently, the volume control has the grounding tabs on the input jacks connected and the grounding tabs on the output jacks connected and then both of these are tied together. Also the wires that shunt voltage to ground via the pots are tied into this mess.
I presume this means that the volume control is actually grounded through the input and output interconnects which attach to a properly grounded amplifier or source."
Weeelll ... sort of!

Your volume control is a shunt attenuator - hence, as you described, "the wires that shunt voltage to ground via the pots are tied" to the input/output RCA ground tags.
As long as this point is attached to the "ground plane" on the Sonic-T amp, all will be well ... even though this ground plane is not "safety earthed" to chassis bcoz you have a plastic case and no power cord!
You said: "I presume that I tie the grounding tabs of the two inputs together. The input grounds are also attached to the circuit board at the appropriate places."
YES.
"Then there is another ground wire that leaves the circuit board, which I imagine connects to the input grounds near the input jacks."
DON'T KNOW ... I can't understand why this ground wire is needed?

Mebbe this gives you the option of attaching the signal ground to chassis ground ... in some environments this can give you a lower noise floor? In which case, you won't need to use it.
"The four output wires go to the respective binding posts, so they don't get tied together, nor are they tied to anything else."
YES
So that leaves only one Q .. which, as I don't know the Sonic-T, I can't help you with. This is ... where does the 0v battery connection attach to? In my battery-powered head amp (2x6v gel-cells in series = 0/+12v), the 0v battery terminal attaches to the PCB ground plane and the +12v battery terminal attaches to the '+' power rail.
I would imagine this should happen with your Sonic-T amp but, if it works off +/- power rails, it will be slightly different.
Regards,
Andy