Volvo - S60 :: 2002 Engine Shaking On Start Up Sometimes

My wife has a 2002 Volvo S60 that we share with ~165,000 miles on it now. In the last 4-5 years or so, it developed this problem of feeling like the whole engine is shaking when we start it up sometimes. It can go days or even a week where we don't have this problem and then all of a sudden it happens one morning or afternoon, or whenever without any rhyme or reason. While it's shaking, the lights on the dash go in and out with the shaking (going between dim and bright). What also happens right before the shaking sometimes is that the engine will rev itself, like its got a mind of its own, sometimes up to 2000 RPMs, like I am pressing on the gas pedal and holding it, but I'm not! It will stay revved up for a good 10 secs or so then come down on its own and proceed to shake. We usually turn the car off, then on again and it goes away. If it doesn't go away, then we repeat (off/on).

We've taken it to the dealer multiple times, including in different cities since we happened to move in the last few years to another city and still, no one can solve the issue. She swears it all started happening after we went a week without the gas tank lid after it got left off by accident after we pumped gas one time. I noticed it was missing definitely within < 7 days and got a replacement right away from the auto store. Not sure if that has anything to do with it?

The car otherwise still runs fine but we just never know when the shaking thing is gonna happen next. It can even happen on the highway when we are driving. You can be trucking along at 70 MPH or so and then ease up on the gas and we'll feel it in the gas pedal and the car...it will jolt and shake a little until we rev the engine above whatever speed we had decelerated to when the shaking started.

When the shaking has happened, I have gotten out of the car to examine under the hood and cannot see anything in particular, the whole thing is just moving like normal and shaking a bit and then eventually if you leave it long enough, it stops on its own.