Doesn't sound like that is normal behavior to me.

What mileage? I just changed engine mounts (2000 C280) at 88m. Indication is rough idle in gear. My other 1999 C280 had mounts changed at around 50m.

Sounds like the car is slipping into gear before you start.

Does the shifter feel normal and stays in position, P,R,N,D? Possible that your shifter bushings are gone and the linkage does not sit correctly. Easy to replace these 2 bushings under the car in the shift linkage.

Better get after this, being a recent used car purchase. Wait too long and it is now your problem.

glenmore
1991 300CE
1999, 2000 C280
2000 C230K