Is it possible to cure some psychological problems through osteopathy treatment? Getting back to shape. Relaxed and so on.

Improve not cure. Certain psychological conditions have physical manifestations in the body especially the paraspinal muscles. Osteopathic treatment can diminish those aspects , relieve stress and help an individual feel more relaxed. They may then be more able to deal with their issues more effectively.
Yes. If by osteopathic treatment you mean exercising regularly, getting to your optimum weight, reducing stress, eating healthy food, then yes, these things can all help to resolve some psychologic problems. Depression and anxiety come to mind in particular. I doubt that osteopathic manipulation can offer much for treating these issues however, unless pain is a large part of the problem, (then yes).