For me, it doesn't look like the car is misfiring or anything. It also doesn't look like the car is having trouble idling. 600-750 rpm is pretty normal idle speed.
My phone speaker isn't great, but in the video it sounds like there's a rumbling type sound when you shift into d or r. To me, that indicates something like a worn or broken engine mount, possibly trans mount. Or that something like the exhaust is hitting somewhere.
I would check these out or have someone inspect the mounts.
I had a 99 e class mercedes. Was smooth in.park and neutral but made a horrible vibration when idling in drive or reverse. Turns out both engine mounts had completely collapsed.
Check them and report back. As soon as I changed both mounts, the car was smooth and all rumbling, noise and vibrations in gear were gone. This could explain why you feel it when shifting into gear, and also when turning on the AC because it puts excess load on the engine and if the mounts are broken/collapsed, you'll feel vibrations and hear noise as well.
