So the audio routing and sync cables are separate things, the sync cables shouldn't go through the mixer at all. As you described it, it sounds correct. Check that each instrument is using the default sync setting (there's a hidden menu on each instrument that you access while holding a button during boot, check the manual for each). You can use either mono (TS) or stereo (TRS) cables for sync, it doesn't matter.

For audio, it is fine to just record a mono signal from each instrument since you don't have enough mixer inputs for full stereo. For that you'll want cables like this:

Hosa CMP-159 3.5 mm TRS to Dual 1/4" TS Stereo Breakout Cable, 10 Feet https://smile.amazon.com/dp/B005HGM1D6/ref=cm_sw_r_sms_apa_i_sw64Eb6ARAN43

Doesn't need to be 10 ft, I would go for 3 ft or something short. You'll connect the black tips only, and leave the red unplugged. That means that you'll record the Left channel only, which is a mono signal.

The above is good for the Volcas, for the Monologue you'll want a cable like this:

1/4 Inch Guitar Instrument Cable 4FT, JSAUX 6.35mm (1/4) TRS to 6.35mm (1/4) TRS Stereo Audio Cable Male to Male Straight-to-Straight for Electric Guitar, bass Guitar, Electric Mandolin-Red https://smile.amazon.com/dp/B081GVRQ6J/ref=cm_sw_r_sms_apa_i_-z64Eb3E4FAVJ

Note that is TS (meaning mono only), not TRS (which could be stereo). That cable will connect the audio output on the monologue (not headphones jack) to a channel on the mixer.

Now with all of the above, your recordings will be left channel only, which is fine as each instrument is mono anyway. In your DAW, simply duplicate each track into the Right channel so that identical sounds come out of each speaker. Or if there's a 'convert mono to stereo' option that should work too. For example, Audacity can do this.