Here is the 220v outlet: It's a three-wire outlet but I was told in another thread that I could use 12-3 wire to also use the line for 110v as well. Do I understand how to wire this correctly: ...

I would run 4 wire cable, since if you ever want to hang a 110v outlet off the line it would be easier with it. (Hot-Hot-Neutral-Ground) and in your case, Neutral would just be left unconnected ...