What is the ruling on someone who prolongs the Duha prayer until the time for the Dhuhr prayer begins, and is his prayer considered valid?

Prayer is prohibited when the sun reaches its zenith (qāʾim al-ẓahīrah), which is a short while before Dhuhr. This is based on the Prophet's (peace be upon him) saying: "Three times رسول الله صلى الله عليه وسلم forbade us from praying in them... and when the sun reaches its zenith."

Therefore, one must finish the Dhuha prayer before the time of prohibition begins. If the time of prohibition begins while one is praying, scholars differ: some say to interrupt it, while others say to complete it lightly. However, continuing the prayer until the time of prohibition ends and Dhuhr begins is not permissible. If you are unsure whether the time of prohibition has begun, the default is that it remains permissible until you know for certain that it has started.