So, I decided to breadboard out my setup for testing, by seeing if I could toggle the remote board via momentary button connected to the Arduino and breadboard, and have the state of that button change the I/O on the Arduino to High for the pin I connected the NPN base to, whose collector and emitter are connected to the remote board HIGH and LOW in the location where the remote's momentary button is connected (see below):
And, I'm proud to say, it works like a charm! Yes! Ignore my solder job, I'm still trying to perfect it.
Gonna try optos next, just for fun, but I'm ready to package this up and integrate it with the rest of my setup, and into my Android app now.
Thanks for those who helped assist / clarify my understanding.