Which is better: to prolong the night prayer, then sleep after sunrise and wake up before Dhuhr only to perform Duha, or not to prolong the night prayer and sleep for a shorter time after sunrise to wake up early and perform Duha and many rak'ahs before Dhuhr?

Night prayer is better than optional daytime prayer, because it involves gathering one's heart and being alone with the Lord, and it brings greater reward and recompense, provided that it does not lead to missing an obligatory duty or harming oneself. One of the best forms of night prayer is for a Muslim to pray part of the night and sleep part of it. Praying the entire night consistently is disliked due to the harm and neglect of rights that it entails.