What should one do who combined Dhuhr and Asr, shortening them, while traveling, then was told that the Dhuhr prayer on Friday is not shortened, so he intended Dhuhr as Asr after finishing it, then prayed Asr with an imam, shortening it, but completed it as four rak'ahs and intended it as Dhuhr?

The claim that shortening the Dhuhr prayer on Friday is invalid is false. The Prophet (peace and blessings be upon him) prayed Dhuhr and Asr together and shortened in Arafat on a Friday. If you prayed Dhuhr shortened and then changed your intention to Asr after finishing it, your Dhuhr prayer is valid and sufficient, and changing your intention to Asr after finishing it is an idle act that has no effect. You must pray Asr because you have not prayed it; it is a debt upon you. However, if you prayed Dhuhr and Asr and then repeated them, your first prayer is sufficient, and there is nothing upon you.