What is the ruling on someone who intended to pray Dhuhr, then realized it was Asr, and what is the ruling on someone who intended to pray Dhuhr while intending to make up for the Asr prayer?

The Dhuhr prayer behind an Imam leading the Asr prayer is valid. This is the view of the Shafi'is and some Hanbalis, in contrast to the Malikis, Hanafis, and some Hanbalis who stipulate that the two prayers must be the same. The determining factor in this case is the intention.