Is ghusl obligatory if one suspects the emission of semen without seeing its trace or remembering a wet dream? And is it necessary to inspect clothing and the private parts every time one wakes up to ensure no semen has been emitted?

There is no need to search or investigate whether semen has exited or not, lest it lead to obsessive doubts (waswasah). The correct approach is to forgo ghusl (definition) unless you are absolutely certain, with definitive conviction, that semen has exited, thereby necessitating ghusl.