If there is no flaking, I have gently wiped the back down with a moist (water) towel and had no issues yet. Would not use anything else. Good saying is the enemy of good enough is better. Not sure what to do with that mud so not going there.

The trim is held on with something similar to black double sided tape. Probably is silicone as RWH said. It is stronger than the white stuff and can be bought at Lowes. I cut the tape with a razor and then pulled the trim off gently but firmly. Be sure not to torq the trim as you pull as it will put pressure on the glass that you don't want. And try your best to hold that glass and not touch the screening. Sounds harder than it is.

If it truly just mud then it should come off with no issues. I would rather have it off then have whatever acids/chemicals in the mud slowly attacking the artwork.
When I say gently, I mean I pretty much just drag the towel over the back. No force on it at all. You will be suprised how much dirt will lift off a seemingly clean backglass.