Is ghusl obligatory if one doubts the emission of semen due to obsessive whispering (waswasa), and what are the implications of that?

The cure for whisperings (waswas) is to ignore them and not pay them any attention. If you doubt that semen has been discharged from you, then abandon this doubt and do not concern yourself with it. Build upon the original state, which is that it has not been discharged, and do not inspect or search. Your prayer is valid in this case, because you have acted upon what Allah has commanded. Ignoring whisperings is an act in accordance with the Shari'ah and pleasing to Allah. As for indulging in them, it pleases Satan and torments you. Therefore, you must ignore them and not care about them.