What should I do if I enter the mosque and find them praying Asr, and I have not yet prayed Dhuhr? Should I pray Asr with them, then pray Dhuhr, or what?

It is permissible for someone who arrives and finds the congregation praying Asr, and he has not yet prayed Dhuhr, to join them with the intention of Dhuhr, and then pray Asr afterwards. The difference in his intention from that of the Imam does not matter, based on the hadith of Mu'adh ibn Jabal. If he joins them with the intention of Asr, he should complete it, then pray Dhuhr, and then repeat Asr to achieve the proper sequence, according to the saying of the Prophet, peace and blessings be upon him: "Whoever forgets a prayer and does not remember it until he is with the Imam, let him pray with the Imam. Then, when he finishes his prayer, let him repeat the prayer he prayed with the Imam."