What is the ruling on a man praying bareheaded, and is it proven that the Prophet, peace and blessings be upon him, did that?

There is no harm in a man praying bareheaded, as the head is not considered awrah (private parts that must be covered) for males. It is, however, good to cover it as an adornment, following the Almighty’s words: "O Children of Adam, take your adornment at every place of prayer." This is particularly emphasized for people of virtue. Scholars have reached a consensus on the validity of the prayer of one who prays bareheaded.