I can be playing music through this, and suddenly, all sound cuts out. Sometimes, if I solo a channel, I can hear that channel.

Unfortunatly, there could be a few different causes for this and without the amp in frony of me it would be impossible to diagnose it. The best bet would be to have a qualified repairman look at it. Here is a link to the Peavey Contact page which has a link to their service centers.
Peavey com

If your mixer has a "Solo" function, that's what you're looking for .

check out your input grounds on your amp

The "rude solo" wil only flash if something is soloed. You need to check again as there is STILL something soloed. The left meter is because that is showing the channel that is being soloed. Once you unsolo everything, the meters wil display full left and right as usual.

With all your plugging in of unsuitable stuff into sockets not designed for them there is a very good chance you'll have blown parts of the mixer. It is apparent you have little idea of what an insert is or the difference between a line and a mic in (or between an in and an out for that matter) I would strongly suggest you stop playing and probably causing damage, read the manual from cover to cover, and if you don't understand it find someone who will help you. I'm sorry but you really are not helping yourself at the moment.

Bummer dude. Sounds like a capacitor or processor problem. Behringer stuff is really not very repairable.
I have had to scrap lots of the stuff over the years.
Great while it works and very affordable

This is likely a problem with the FX in the unit or the power to it. Does the FX display act correctly? Check that the headphone source "Main Mix" is the ONLY button down for the headphones. If not, make it so. If the clicking is still there, try the headphones in the aux send jacks... see if EITHER left or right can hear the clicking. Back in the phone jack, scan across the SOLO channel buttons down one at a time. Is there clicking? If not, the problem involves the mixing buss or further down the line. Make sure the SOLO buttons at the FX Aux sends are up.
What effect does the MODE button have in SOLO and in PFL? Is the SOLO LED off? Is the number in the FX window SOLIDLY on and not flashing?

Who would have any idea of what make or model device you have? Sounds like some brand of organ. We are not mind readers on this forum.

When it is working, wiggle the channel volume control WITHOUT rotating. If the sound cuts out, likely the soldering of the pot leads is broken where it attaches to the board. Try resoldering the connections.

SOMETIMES the control is damaged in which case it must be replaced.

Here is the manual if you don't have one:http://www.mackie.com/pdf/archive/cr1604_om.pdf. There are solo buttons for each channel. There is also an auxiliary return solo button. There is also a solo to main button. These buttons are all inter related. Is your main output mute button in or out? Read the manual carefully as there are a myriad of combinations you can use. Have a look at the trouble shooting guide on page 21. If you can't find a solution try contacting Mackie. Let me know if this helps.