Does the Witr prayer performed just before dawn on Sunday count for Saturday or Sunday, and what is the ruling on repeating the Witr prayer in a single night with the intention of it being for the following day?

The night follows the day after it. So, the night of Friday begins with the sunset of Thursday and continues until dawn. The time for Witr prayer begins after the 'Isha prayer and ends with the true dawn. Therefore, if you pray Witr before dawn on Sunday, you have prayed it during a part of Sunday night. As for repeating the Witr prayer twice, for example, praying it on Thursday night and then again before the dawn of Thursday, this is forbidden by Islamic law.