I put the "head banger" at the end of my post because that's probably how you feel .... hang in there.
The basic job of the MPEM is to supply voltage to the different components of the ski. The basic job of the ECU is to manage engine operation (sensor inputs, injectors, coils, etc.) The ECU fires the coils AND the injectors by pulsing their ground circuits. In your case, the coils are receiving a constant ground from a shorted ECU which is causing them to get hot. The ECU isn't totally shorted just partially shorted. That's why some components are still working correctly (injectors) and some not ( coils).
It would be a big help if you would get a service manual. It shows overviews of the different systems and how they relate to each other. Has wiring diagrams too.
In my opinion, you're at a point where you need to replace the ECU, necessary coils and then see what you've got. Make sure the harness isn't damaged somewhere first.
Keep in mind that the DESS key "reports" to the ECU and therefore will have to be programmed to the new ECU. Good Luck.