I'm just wondering if I need to pick up speaker cable, or if I can send the amplified signal from my amp to my cab via a standard mic/instrument cable.

It's for recording purposes, so I want to have the amp in the control room and send the signal through my patchbay into my live room, but I don't want to break anything or start any fires.
Speaker cable, definitely. Especially over long runs. Instrument cable isn't meant to handle the higher current and the conductor may burn out, which can mess up your amp in a hurry.
Yeah, there's no point in using my patchbay anyways, as I could only send it out a standard mic cable with the way I have it set up. I have an old semi-plugged hole from before I set up my snake that I can use to run a speaker cable.