...Is it possible to conceive a turbo jet that works as such but a part of its power is derived to a shaft that moves a propeler? Or a turboprop whose exhaust thrust is relevant (more than marginal)?
The key word there is "marginal". Exhaust efflux from a turboprop generally produces less that 10% of the total thrust. (Exhaust thrust is also further reduced by onboard systems tapping bleed air from the compressor, etc.) You might imagine gearing a turbine in favour of exhaust thrust but you'd probably be better off going with 'mixed-power' - ie: turboprop + turbojet/fan.
BTW, exhaust thrust also explains why you see power ratings listed other than shp - like ehp (equivalent horsepower) or, less frequently, eshp (equivalent shaft horsepower). Shaft horsepower refers only to the thrust generated by the propeller. But ehp and eshp combine the thrust generated by both the propeller and the exhaust.
All that said, you'd need an exhaust pipe for a turboprop anyway ... so your
MustanGina's tailpipe is already good to go
