Is money earned from winning an e-gaming competition—where a participation fee is paid to cover the venue's rent, unrelated to prizes—considered gambling and thus forbidden? And is it permissible to participate in such competitions?

This competition involves gambling because the participant pays a sum for entry, and their situation oscillates between gain and loss. They might win and gain more than they paid, or they might not win and lose what they paid. This is precisely gambling (al-maysir and al-qimar). Furthermore, it is not permissible to take prizes in these games because the Prophet, peace and blessings be upon him, said: "There is no prize money except for races involving camels, arrows, or horses." These games do not fall under these three categories. Therefore, it is not permissible to participate in them with the aim of obtaining prizes. Your friend must repent and dispose of the acquired money by spending it on charitable causes.