I need help with understanding a turbocharger on exhaust side:

1) Exhaust gas is not being compressed, it push the turbine, then exit. Correct or wrong?

2) Exhaust gas hottest from cylinder, cooler after turbine? Yes or no. If yes, how much cooler?

 

1. The turbine side of the turbo does not compress the exhaust gas. However the exhaust gasses before the turbo will be under pressure as the turbine is a restriction to flow.

2. Yes. How much cooler is dependent on the efficiency of the turbine you use. Part of the energy the turbine is extracting is in the form of heat.