What is the ruling regarding someone who prayed standing on his shoes (meaning, with his feet on top of the shoes) on Friday due to intense heat and the absence of anything to prostrate upon, while using his skullcap for prostration?

If the sandals are pure, then your prayer with your feet on them is valid and sufficient. The permissibility of praying in sandals after ascertaining their purity has been mentioned. As for prostrating on a prayer cap, the prayer is valid according to the majority of scholars.