I never used your mixer but I read the manual, and looked into it.
What I'm guessing that it does is mix the Line Input with the 2 track Input. The line input can be switched between the RCA inputs or the USB input. Therefore you have either:
- RCA Line Input + 2 Track RCA Input or
- USB Input + 2 track RCA Input
You mentioned that when hitting the red button you can hear the game PC or stream PC. This is because the Game PC is connect to the RCA Line input, and the stream PC is connected via USB. Your switching between these 2. The connection to the 2 track input from the stream PC is doing absolutely nothing since the stream PC is outputting audio via the USB. You need to set windows to output sound to either your soundcard/onboard sound or USB. Do this in control panel under sound.
Now here is what to do. Disconnect the game PC for now. Hook the USB to the stream pc (required to get the input form the mic). Connect the soundcard to the RCA line inputs. Now switch between the 2 using the red switch. It will only work on USB for now, so switch windows to output to USB and Soundcard as you hit the red button. Listen to the difference and determine which sounds better. I'm guessing the USB will sound better and be quieter with less hiss since your suing the Behringer DAC with digital input. While you rin control panel under sound you can check under recording device to make sure your mic is set to USB which is probably already is.
Ok, so lets say you determine USB is the best. Disconnect the RCA cables and set the red switch to USB. If you prefer the RCA connection leave it connection, set windows to output via soundcard not USB, and set the red switch so you hear from the RCA. The USB must remain installed for power, and to get the mic signal to your computer.
Now connect the game computer to the 2 track. It'll mix with either the USB or the RCA Line input from the stream computer, whichever you determined sounds better.
The problem you have now is the game computer is connected to the line input. Your stream computer is outputting via USB. You cannot mix Line In + USB. Only switch between the 2. You need to move the game computer over to Track 2, and set to USB to mix the 2 and eliminate the RCA from the stream computer. Or plug the RCA from stream to Line In if you determine it sounds better and set windows to output via the soundcard.
Windows cannot output to 2 devices simultaneously.
It's hard to explain without writing a lot, but I hope you can follow along. If you have any questions feel free to ask. Also make sure both buttons are set to send to headphones so you hear both computers and not just one. Once working mix the 2 computers to balance the levels to your liking.