Does one have to perform ghusl if one has sexual fantasies and feels pleasure during sleep, and do these fantasies invalidate the fast and the prayer?

What you did is not masturbation. The cure for obsessive thoughts (waswas) is to ignore them. You are not required to perform ghusl (ritual purification bath) merely based on doubt. Rather, it becomes obligatory if you are absolutely certain that semen, which necessitates ghusl, has been discharged. If you doubt whether it is semen or something else, you have a choice. You should abandon these obsessive thoughts and not pay any attention to them.