Solved

How to train crew members? Doctor-Designer-Technical Director

I bought a doctor 2 seasons before. But he just loosing stars slowly, and not improving at all. How can I train my Doctor? (or Designer/TechDirector)

you cant train them! need to sign new ones

Do they ever improve in stars with experience and morale increases? I haven't seen it yet but I have seen someone with staff above their level.

The stars decrease because your manager level is going up, Zsolt.

Stars are used to quickly indicate the level of things in relation to your manager level.

I've forgotten, I think experience might, but since these stats don't play a role in the level of the staff currently I've not paid any attention.