"How do I stay on the mat"

I think you'll just get used to it. There really isn't any magic there.

"My SAR pack will have the Nemo Switchback and the Sleepingo Inflatable Pad"

Note that the pad you link is a 32F pad. Presumably because it it mostly an air mattress at the expense of having much insulation. Air mattresses and cold do not go together. You should do another 18F test run with that to see if it works for you. It wouldn't for me, I'm a cold sleeper.

"Do you put your pillow inside the sleeping bag or underneath?"

I put mine in the bag.

"Any tips for making the material of the sleeping bag a little warmer? When I would adjust my body position, I'd end up touching a cold part of the bag (inside). "

Get a bag with a colder rating. Yours is a 20F bag and you were camping at 18F. That's right on the razors edge. Most people, including me, want a sleep system to be rated colder than the actual temps. For me, a cold sleeper, if it's 18F I'm using my -40F system with an overbag.