If the ETA has not yet been rebuilt, that is most likely the root cause of the idling issues. Do you know the age of the ETA (date code), and if it's original?
The instrument cluster is discrete and completely separated from all engine management systems. If you lost all instruments, that indicates a power supply issue - need to inspect/clean all fuse ends/sockets, check battery terminals, maybe the ignition switch, check fuses in the BM/GM, etc. Check the FSM schematics for the instruments if needed. Loss of instruments will not be related to ETA/E-GAS, but the ABS/ASR lights would come on if the module lost power.
124-545-26-32 is correct for an early 500E outside of USA. An alternate number would be 124-545-44-32, this is for later 036 but is interchangeable. I don't know why the North American cars have different E-GAS numbers, how they are different, or if these are interchangeable with ROW modules.
1. I haven't seen that before. Not sure how to clean it. Also don't know if it's affecting the E-GAS functionality or if it's strictly cosmetic.
2. VERY few places repair, or claim to repair, E-GAS modules. Beckmann in USA said they could do it, but cost was over $1k last I checked.
3. eBay would be the primary alternate source, just make sure the seller accepts returns, or better yet has tested the unit and guarantees it's functioning correctly. Some eBay sellers claim a module is good/tested when they've done absolutely nothing to check it.
Before shelling out $$$$ for E-GAS repair/replacement, try sourcing one from a 140 chassis as mentioned previously and see if that magically cures all your woes. Once you confirm the E-GAS is actually faulty, then worry about repair/replacement. I have a NOS unit available but it's for North American cars, I'm not certain what the difference is with ROW.