Temp reading on dashboard and vibrating sound while stopped after axle change. VW Passat

I drive a 2003 VW Passat StationWagon.

The mechanic changed the axle abt a month ago and I can now feel/hear a vibrating sound when I stop at a light. My husband says this is lesser 1) when he puts it in neutral 2) when it is warmer outside or after riding for a while. I agree with 1) but not 2). The mechanic says this will happen and we should wait a bit for this to lessen. Is this so?

In addition now, we notice that the temperature guage on the dashboard is not showing a reading. Not sure it is related to the axle change, since we did not pay attention to it that time. We got the mechanic to take a look and he says the engine is heating up fine (the heater comes on fine, though I think it is not as powerful as before), and he feels it may be bcos the thermostat is exposed outside. He asked us to wait for a couple of weeks to see if it corrects itself. I am concerned about this. Please advice.

The info is sparse but I don’t see the axle change having anything to do with the current problems.
A vibration at idle only, with the car stationary, could point to any one of a number of things including something so small as the engine idle speed being too low.

This sounds like an engine driveability problem to me but there’s simply not enough known about the car to be able to even make a wild guess as to what.

Thanks. The car has 75K miles on it. It has a 4G turbo cylinder. Sorry, I’m not a car person, so please let me know what other info would be helpful, and I’ll get it for you.

if during the axle change the computers temp sensor was damaged or disconnected this could cause both problems . not only could it throw the temp reading but it will affect the cars air fuel mixture causing a rough idle. another thought is that one or more engine mounts could have been moved to do the job and not been re-installed correctly causing a new vibration that was’nt there before. last thing A MACHINE CAN NOT FIX ITSELF at least not yet