well, for one thing, i dont believe it is a jet turbine. It appears that the thrust is solely produced by a giant propeller.

The gas generator that runs the big fan is probably ventilated well enough not to burn the pilot.

The article talks about it getting up to 300C. Regardless of how it works that is a little concerning.

I'd be more concerned about landing it. There's no cage around the pilot and it must weigh quite a bit, if for whatever reason you lost thrust for a moment during a descent you'd crush your lower body as you fell to the ground and the machine strapped to your back fell too.

Or even the straps of the harness restricting circulation to the pilots legs leaving them numb at landing. Ever try to walk with your foot "asleep".

I'm not an airplane engine expert, but they show a compressor and turbine blade. Would that make it a jet turbine?

The term "jet" (or turbojet) tends to mean that the thrust comes from the exhaust of the engine, and all the turbine's power is put back into the compressor. In a gas turbine, the turbine captures more energy from the combustion to drive a driveshaft, and the excess exhaust is dumped (or in this case, used as additional thrust for control).

with 25m3 (800 cubic feet) per second flow of air (at the speed of 100m/s) behind your back ? (that's a mass flow and speed he needs to get through his 0.6m fan to get 250Kg of force for 120Kg device + 80 Kg pilot) Letting a bit of it out to caress your back doesn't seem to be a problem.