OK, so I did the oil and filter change, air filter change and full diagnostic of the vehicle. The diagnostics guy said he did some kind of software update too (although I have no idea what kind). He said the CEL might come on, because of the ambient air temperature sensor misreading temperature (it thinks its 109F all the time, sometimes even goes to 110F. in reality its 10 degrees Celsius, that should be about 50F at most), although I have no idea what connection does it have with the engine. The Torque app on my phone reads air intake temperature perfectly, so why would engine control module care about the ambient temperature, is a real puzzle to me.
Somehow vibrations went away too. It still vibrates, but it is muted by very large margin. It is hard to pinpoint why, because we changed fluid in engine and gearbox, changed the spark plugs, air filter and drove about 200 miles on premium gasoline (just to be sure that the fuel was correct), also I have used some fuel additive from LiquiMolly that said it would clean the injectors (which I honestly doubt it, but still, how bad can it be?).
When changing oil, we discovered that the gearbox mount is shot. I somehow found the part but did not have time to change it. Motor mounts seem OK. The testing procedure was applying gas with handbrake on in D and R and in both cases engine did not budge. It seems, previous two mechanics were correct when saying the mounts are good, but it may be that the mounts are just old and transmit more vibration that they should. Anyway, there are none for sale at any shop or garage, so I don't know if it is worth to purchase them on Ebay or any USA/UK store and then bother with custom clearances and stuff like that.
Another thing is that we could not find the Power Steering Motor. There does not seem to be a part like that on a car. This is very interesting, because the steering wheel is heavier when the car is running and lighter when the car is turned off. I have never encountered anything like that and I am very confused. I thought it was an issue with resistance of a bad electric motor, but there isn't one, so... Any suggestions of this front would be much appreciated.