For example in the picture below we can see that the turbine of the compressor is just behind the combustion chamber and gets the hottest gas and i think therefore can extract the most energy? The power turbine comes after the compressor turbine but power turbine is the turbine that propels the vehicle. Why is the power turbine placed after compressor turbine if it is the one propelling the vehicle, shouldn't it be placed first so that it gets the hottest gas and extracts the most energy and propels the vehicle forward with more torque?
The compressor-turbine spool, the "gas generator" in turboprop/turboshaft lingo, or "core engine", in a turbofan, has to come ahead of the power turbine because the turbine has to be directly connected to its compressor to maintain the combustion cycle.
The power turbine needs to go after, its shaft either running inside the spool of the gas generator out the front, or out the other way (like a PT-6, similar to your diagram).
To put the power turbine ahead of the compressor turbine, you'd have to transmit its power through some transmission method at the outer rim of the turbine, maybe using a ring gear on the outside of the power turbine.
Aside from flow issues likely to have to be overcome when you are extracting energy ahead of the compressor turbine, you would have an unwieldy transmission structure on the outside of the engine, where a pinion gear would mesh with the outside of the power turbine. Just thinking about it sounds like a mechanical nightmare.
So much easier to stick the power turbine in the exhaust of the gas generator, like putting a windmill behind a regular turbojet's exhaust.
In any case, the energy required to compress the incoming flow has priority over energy being used to make torque, so that's another reason to put the power turbine after.
The "compressor turbine" is the turbine (or group of turbines) which extract from the exhaust enough energy to drive the compressor stages, nothing more.
The remaining energy possessed by the exhaust after the "compressor turbine" can be used in two ways:
we add another turbine (called "free power turbine") which we connect to the propeller (in a propeller airplane) or to the rotor (in a helicopter) or to an electrical engine/generator (for ground applications); in total some 80% of the energy is converted in torque on the shaft;
if the jet engine is used to push (thrust) the aircraft then no "free power turbine" is needed and the whole exhaust energy goes into "pushing" the airplane.