How to set it up on a non power speaker

On the back panel there is a section labelled Patch Bay
There are 2 small switches which assign each Power Amp to either Monitor or Main Output.
It would seem you have them both set the same at present. Set one to Monitor and one to Main, and use the corresponding Power Amp outputs direct to speakers.

Note you can send signal to external amps or powered speakers from the preamp outputs. This is really handy.
Download and print the manual from https://www.manualslib.com/manual/90516/Kustom-Kpm-8420t.html
and keep it with the mixer.

( Of course don't send speaker outputs to other amps or powered speakers )

Is your graphic eq assigned to main and is kit set correctly?

Then you either have a blown amp or speaker. Turn the amp off and move the wires from the speaker on the working side to the other speaker. If the other speaker works, you have a dead amp channel. if it does not, move the wires from the speaker on the non-working side to the other speaker. If that works, you have a dead speaker. Dead amp channel means take it to the shop. Dead speaker means buy a new one from Line 6 and install.

The pop itself isn't necessarily a bad thing, but a burn smell and the fact that it's only outputting a crackle likely means that the built-in amplifier circuitry has been overloaded. This can happen a few ways:
1) Faulty power, either a bad cord or a bad outlet
2) Plugging the output of a power amplifier into this speaker - because it has a built in amplifier, this speaker only wants a non-powered line input
3) If the Thru output is (accidentally) plugged into anything that provides phantom power, that could overload the circuitry.
4) A short due to liquid or moisture contact - these cabinets are most certainly not waterproof
In any case, the amplifier part of the circuitry IS replaceable. If you're not comfortable buying one and replacing it yourself (I'd recommend eBay), take it to any shop that sells similar items. If you're extremely lucky, it was just a fuse that needs to be replaced, but chances are you'll need to replace the whole amp.

In a word, NO. The line out is line level audio... it has NO power amp and cannot drive a non-powered speaker. I have both of these cabinets myself. You need to use an amp with the non-powered speaker. When using an external amp you will need to make sure you phase the two speakers correctely... you could just reverse the speaker leads of the non-powered one if needed.

Balanced audio interconnects (either XLR or TRS cables from the mixer and then balanced audio interconnects from the amp daisy chain inputs to the existing speaker inputs. If the amps do not have linked input jacks, then use TRS type WYE connectors at the inputs. Then use speaker rated cables from the amps to the passive speakers.

You didn't mention what brand powered mixer you have, HOWEVER MOST powered mixers also have a line level monitor and main output jacks. These are almost always 1/4 inch jacks. In any case you should be using BALANCED lines to the powered speakers using either XLR or TRS connectors on the ends. Please tell us what powered mixer you have so we can give you better answer regarding which connections to use. IF for some reason your mixer does NOT have line level outputs, you can use a high level Direct Box to condition the powered outputs to be sent to the powered speakers..
Also make sure ALL your interconnected equipment is powered by the same receptacle or source to avoid damaging due to ground bounce on utility lines and circuits. This often means running an extension cord from the stage where the powered speakers are along with a snake of the audio cables.

You haven't mentioned what model and manufacturer the mixer is. I will have to guess and say that the monitor out on the board isn't providing signal to the powered monitor or the monitor out on your board is powered and is now damaged by plugging a powered monitor into it. Carefully reread your manual and determine if the monitor out on your mixer is powered or line level. Reread also and make sure that the signal you are monitoring is assigned to monitor out, levelled correctly etc. If you find that the monitor out is passive (line level) try hooking it up to an external power amp and speaker. A guitar amp will do for testing purposes. Start with a low input volume on the amp. Also make sure that if it is passive to use a shielded cable from monitor out to input. If your board has powered monitor outs you can use a speaker cable from monitor out to speaker input (non powered monitor speaker).

Troubleshooting is always a "process of elimination." Have you tried hooking up your Peavey powered mixer to another set of speakers? How about connecting the speakers to another audio system? Your term "very little sound" could mean audio input problems, mixer/amp settings or failure and possibly speaker damage (internal passive crossovers). Please be more specific.

Yep, chalk that one up to poor design...The amplifiers are going out of balance during power up and down. Two seconds is pretty long... Try turning down the gain during power on and see if that affects it.

The pop on turning off is normal... None of these have speaker disconnecting hardware that is necessary for very high power amps. The biggies have hardware that checks amplifier is balanced before connecting the speakers and a relay also cuts the speakers immediatley on power down. As power goes up and down the circuitry comes to equilibrium and until it does, strange things happen.

I would investigate the CONDITION of the filter caps for the main power because if they age to a higher internal resistance that COULD cause the high freq oscillation on power up... Two seconds is MUCH too long.

There should also be some non-electrolytic caps NEAR the power output transistors to handle high frequency supply power bypassing.

Check the power bypasses that service the preamp stages. Look for pregnant electrolytic caps that have failed.

Ultimately, you MAY need an oscilloscope to isolate the high freq problem.