What is the ruling on fasting Friday, the Days of Tashreeq, or the Day of Doubt for someone who observes the Fast of David? And are there any regulations for this type of fasting?

It is disliked to fast on Fridays and on the Day of Doubt if they are singled out for fasting. It is forbidden to fast on the Days of Tashriq, except for a mutamatti' (one performing Tamattu' Hajj) or a qarin (one performing Qiran Hajj) who cannot find an offering. The fast of Dawud is to fast one day and break the fast one day, and it is the best of voluntary fasts. If his fasting day coincides with a Friday or the Day of Doubt, there is no harm or dislike, because this falls under the category of coincidence, not intention. However, if Dawud's fast coincides with the Days of Tashriq, then breaking the fast is obligatory, and fasting is not permissible except for those to whom the Sharia has granted a concession.