After only an additional day of investigating, I have the second monitor hooked up and functional!
Entirely as expected, all I needed was a different cord because another HDMI cable and splitter was not cutting it giving me the results that I wanted in order to be able to game and stream on a second monitor.
I’ve set up almost everything that I want (and, in the case of streaming, need) on that second monitor, and am deliberating upgrading our computer’s webcam after so long. I’m also considering getting a streaming keypad to be able to queue things up even more easily and with no more than single button presses. That should cover just about everything “professional” I’ll need to stream, and stream well, at this point in time!
With luck, I should be able to take pictures of these things as they come along over the weeks and months!