The U2 ICE2A165 chip should be replaced by an ICE2A265 (just as cheap). It's a standard 8-pin IC; if I was working on that board, I'd pull the broken chip, insert an 8-pin IC socket and then use the upgraded chip (it has better thermal properties). Talk to another repair man - this doesn't seem like a challenging or expensive fix.
This is generally expected behaviour, but that does depend on your amp. Some amps are designed to give you more clean headroom, but you don't have much headroom in a small amp.
The answer is to not crank the volume up as high. If you need more volume, get a higher wattage amp.
If you want the real simple solution; record into your phone, but put the phone further away from yourself so it isn't biased on where it's pointing.
Use the condenser mic you have, and hook it to your computer. Record the sound there.
Also record the sound (and video) on your phone. Transfer that file to the computer.
Use music software to ...
There is only one channel available on the powered speaker. So doing anything 'stereo' is a waste. You have L/mono out of the drum brain. That sums it all together. Use that, and plug into one channel of the Behringer. One signal in, one signal out. Simple, and using the simplest plug arrangement - mono to mono.
To make it sound really good, get another ...
David Gilmour's tone is described on various sites, but the basic core of it is a clean tube amp with vintage pickups. His amps (Hiwatts) also have a lower mid range, so you can emulate that by turning down your mids, but even on the Hiwatts Gilmour has his mids at 40%, bass at 50% and treble up towards 60%.
If you have a delay pedal, that would help get you ...
You may use either a TS or TRS to the line/guitar input of the Bose either way. The input is probably TS since it is set up to accept an instrument level or Line level and nothing indicates that it is balanced input.
If you send a balanced signal through a TRS jack to a TS input, the second signal will be sent to ground and not used.
If the Bose does accept ...
Case closed. I contacted a local asbestos abatement company and the manager says asbestos was at one time used as fire retardant in some electronics. Anyway, said company is disposing of it. Thanks all.
Since you have one speaker and the Behringer L output doubles as a mono output (L/mono) use choice “2”. All you need is a single mono 1/4” instrument cable. The drum machine will output a mono signal if nothing is connected to the right output and a TRS type cable is not necessary since it is a mono output.