Yes, I am insured with the operator's insurance. That is first thing I checked!
I don't understand you other question - I have a CPL so I can fly for hire or reward, and I could (if I wished) invoice for my services as a pilot to the passengers. I just choose not to do so, as I enjoy the flying.
All I can suggest is, as I have said before and Bravo has implied, is that you should not arrange the aircraft hire.
Exactly, that is the key point. Incidentally I also learnt today that if somebody wishes to hire an aircraft in this fashion it is considered private hire and they are not restriced by JAR/OPS. So they can, for example hire a SEP, then hire a pilot to fly it.