Is it necessary to inspect for semen upon waking after a wet dream, and how can it be distinguished from other secretions, and is its exit limited to moments of intense lust?

One must turn away from misgivings and pay no attention to them. Ghusl is not required when in doubt about the emission of semen. Rather, certain conviction of its emission is necessary. If a questionable fluid is discharged, one can choose a ruling for it. It is advised to refer to Fatwa No. 118778 and its contained references to learn about the characteristics of semen and the difference between it and other discharges.