The schematic could be improved with a couple of arrows indicating direction of flow, but it's not too bad. Yes, bleed air from the engines can be a lot higher than 32 psi, but that's the pressure delivered (via regulating valves) into the aircraft pneumatics.
First, it's cooled by the primary heat exchanger, then compressed to 50 psi by the right-hand rotor, which is a compressor, NOT a turbine. As it's compressed, of course it's heated (adiabatically) as well.
Then it's cooled some more by the secondary heat exchanger, and routed to the turbine. There it gives up some pressure and temperature as shaft energy is extracted to drive the compressor.
The air has now cooled to the point it's saturated with moisture, hence the water separator.
Finally some hot air is mixed in to bring it to cabin temperature.