UPDATE
Earlier:
I had some issue with RPM drop and stay low when rolling -- to address this, I cleaned the throttle body and MAF sensor. This fixed this issue.
Later, I started noticing vibration in D when stopped. I decided to not follow the advice posted in this thread, and I didn't play with clutch engage via obd2.
Recently, I went to dealership, and paid $300 for diagnostics.
At the dealership, the technician found stored code P0661. According to the logs/freeze frame, this code appeared right after I decided to give up trying to figure out what was wrong with that vibration. This code didn't trigger the check engine light or other lights.
As per technician report, one solenoid was stuck in open position, and I guess, the other one was kind of failing.
So, I replaced the solenoid valve LF82-18-740 and cleared the code.
Here's a great video guide on this fix:
The vibration in D when stopped is gone, well, almost gone.
Now, if I stop in D, the rpm might occasionally drop to 600 and below, the vibration happens, but then the engine adjusts the rpm to 620-660, where there's no vibration.
Previously, it would just drop flat to 600 and vibrate.
I guess, it will take some more time for ECM to relearn. Normal RPM idle when no abnormal vibration is noticeable is around 650, I think.
I'll update this thread with MPG observation later, but overall, I consider this issue as resolved.