Will a male, 13 years of age, be held accountable for his actions after semen is discharged through masturbation, and before a wet dream? And what is the method for performing ghusl and repentance in this case?

Masturbation is forbidden (haram). The emission of semen, regardless of the method, is an indication of puberty and accountability, and ghusl (ritual bath) becomes obligatory after its emission. As for repentance from masturbation, it is achieved by refraining from the act, regretting it, and resolving not to return to it.