Is it obligatory for someone who slept through Dhuhr prayer and woke up just before Asr to pray Dhuhr at home and then go to the mosque to pray Dhuhr again with the congregation?

If you wake up before the Asr prayer, perform ablution, and pray Dhuhr, then you have prayed it within its time. However, if you hear the Adhan for Asr from a trustworthy person knowledgeable of the prayer times, and you have not yet prayed Dhuhr, then it becomes a missed prayer (fā’itah) according to the majority of scholars. It is recommended to perform the missed prayer in congregation if possible, and there is no obligation to repeat its performance more than once.