OK. I think I'll try to answer my own question. I hope this helps others.
There were two issues that are somewhat related but separate. I'll address both.
1) The mix knob.
Since there is no physical mix knob on the Studio 1810, I've found that you must utilize a software volume controller. There are several ways I figured out how to do this.
a) Just turn down the Master Fader. OK, not great but works fine. You run the risk of opening a new song, forget to pull down the master fader, and then blow the ears off of your headphone listeners. That's not fun. You also have to pull it back to 0 to continue mixing.
b) Better solution. - Create a new output on channels 3/4 in the Song Options Audio screen. Mark this output as a Cue Mix. Now, you have an independent output level AND independent channel volumes that you can control, so you don't have to mess with any channel sliders you may have set already and don't want to change. This option only works well, though, if you change the way that the Universal Controller software is set up by default (that's problem #2 that I asked about and will answer below)
With option B, you can hardwire the 3/4 output to whatever channel on your board you want. I piped it into an open stereo channel I have. Now, if I want us all to hear the Cue Mix, I just turn off my main input and turn on the Cue Mix input and there it is!
#2) The problem was output 3/4 being summed into the main output. This happens due to the Universal Controller software. It's a little weird to figure out. But when you check on the big 3/4 output buttons on the right, you get a whole new set of channel sliders that are yellow as opposed to the Main out White. You have to ensure that your 3/4 outs are not going into your Main out and that your Main out are not going into your 3/4 outs. Crazy that it's defaulted that way.
Once those are muted correctly, they act totally independent of each other.