Hi Chris. Yes, back to the grind.
I would replace the cord or the end to get a good ground. GFCI should help for protection, but ground plug is best.
The stc-1000 in your instance will be powered by the plug going to the wall and then will switch the 12v dc from the wall wart then go to the SSR. It isn't switching the 110v that will run the heating element. So they won't share a 110v cord.
Like I said the STC-1000 isn't so essential to ground. Unless the controller falls into the water it has very little risk of shock.
That is the way the stc is intended to be used. although the addition of the SSR increases the current load that it can control.
Once you start wiring the three devices together i think it will prove clear how they are separated. Get started and follow the drawing i sent you and if you get stumped, send me a snapshot of what you have so far.
I see what you are asking. if you connect the power from the plug to the stc-1000 and then carry the ground and common over to the receptical. (Second hole in your box correct) And then take the hot lead 110v to one side of SSR and then from other side of SSR to the hot screw on the receptical, you will effectively create a regulated/switched plug that you can then plug any heater into. Then use an old cord to go from plug to your element and attach ground to it.
You got it. Yes to two cuttouts.
You may be better off just making a big loop of your gruond wire and puting it under the stainless nut. Soldering it to it won't likely hold