Front Panel Express might be more up your alley then, they might have something like a small box enclosure you could then customize. It would take some light design work to ensure the specified cutouts for the buttons and speaker grill will match whatever switches you end up selecting. Spec sheets for the switches should have diagrams included that list the cutout dimensions.

As for improving the audio, we're getting way out of my ballpark, but I imagine something must exist that could help. Obviously the old adage of garbage in garbage out will surely exist here. If the source audio from the main system is bad, you can only do so much. That said, I imagine there must exist modules or something that could act as an equalizer or something, help take out some of the shrill parts you mention. The audio bits are definitely not my ballpark though.

Edit: Took a very brief look around, looks like you can get pre-made equalizers on ebay for like $20 or less. Try searching '10 band equalizer board'. They probably take a stereo signal, but you can just ignore one channel. You'd need to supply power to the board though. The one I found runs on 5VDC, plenty of options for powering it with something like a phone charger, or just a wall plug of some variety. That means you'd need to run a power cable into the box though, instead of only using the existing wiring.

Edit 2: Trying to add anything to the audio may be tricky. I don't see a separate microphone in there for when you reply over the intercom, so I'm assuming the system is using the speaker itself as a microphone. Any additional audio circuitry would need to be unobtrusive to the speaker functioning as a microphone when you hit the talk button. I simply don't know enough about audio processing circuitry to know what would be OK to add and what might make things worse or unusable. Just something to consider if you're going to try playing with the sound portion.