In the notes it was correct in one place and wrong in the other. I have corrected it in the notes. The only conditions necessary for the stagnation temperature to be a constant are that the process be adiabatic, steady, and involve no external work. When we get to stagnation pressure and its relationship to stagnation temperature it will be necessary to assume a quasi-static process, but it is not necessary to do this for stagnation temperature. It doesn't mean that the flow will pass through such a process, but often it does.
Flow always stagnates on bodies.
The higher the pressure, the more potential there is to get work out of it. You have a right to be. Let me hold off explaining more of it for now since we have a whole lecture devoted to it on Friday. The last PRS question was confusing. Read the notes it should help.