Good news: I inadvertently had wired the motor on the ETA backwards, and now after correcting it, the car starts and runs without limp mode. I ran it for 30 minutes without stalling. It now buzzes as it should with the ignition on. I initially got the idea from this thread: Connector for ETA - and rewire

Victor of http://www.restoreyourmercedes.com/index.html was as kind as to confirm which pin goes to which terminal (left, right) inside the ETA. They are not marked.

Bad news: Once the car is warm, with A/C on, placing the transmission in drive or reverse makes the engine idle below 500 rpm, and almost stall. I have to quickly switch to neutral and press the gas pedal to save it. Also, there is a vibration around 1,700 rpm. It seems there is a missing cylinder.

I am planning to do the following:
Restrict air to the MAF sensor with hand on top, to see if system compensates by opening the throttle
Disconnect the MAF sensor (I just cleaned it), and run engine, to see if it runs better
Listen to each injector with a stethoscope, to verify they are all firing
Plug each vacuum port on intake, to ensure there is no vacuum leak
Erase and read the codes
Compression test, but I think I would have worse symptoms with a stretched or jumped timing chain

Should I test anything else?

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2004 E500 4Matic 138,000 miles