What is the ruling on performing ghusl after finding traces of semen on the bed, and should the Fajr prayer be repeated, given the uncertainty of whether the semen was discharged before or after Fajr?

The presence of semen is attributed to the last sleep the person had on the bed or in the garment where it was found. So, if a person finds dry semen and does not know when it occurred, he must perform ghusl (ritual bath) and repeat the prayers he performed since the last sleep he had in that garment. What the questioner did by performing ghusl and wudu (ablution) for Dhuhr prayer is correct, and there is no consideration for the possibility of semen being present before Fajr, because the principle is that an incident is attributed to its nearest time.