The picture you posted looks like it could be the ECM. the idle control is located under the carpet on the passenger side. It looks just like the fuel relay and is pn 004 545 11 32. The ICM sends a control current to the ICV to control idle speed.Check the current. When you pulled the plug off the ICV and the rpm went up that shows both valve and control are working. To determine if they are working properly you study control current.

To answer your second question I read on posts here you usually can't tell by looking at the boards whats wrong, but if you remelt all solder joints (some add a little solder some don't) sometimes it fixes the problem.

I don't recommend putting a solder iron to the ECM.

When driving (not in idle) how is the engine?