What is the legal evidence for shaking water off the hands before wiping the head during Wudu (ablution)? Is there any Sunnah regarding it? And is it permissible to do so according to the four schools of thought?

There is no evidence that prohibits shaking one's hands before wiping the head in ablution (wudu'), and the default rule is permissibility. The intention behind shaking the hands might be to reduce the water so that the head is wiped rather than washed. It is sufficient for there to be dampness on the hand after shaking it to wipe the head. Scholars have mentioned that it is recommended when wiping the head to wet the hands, then pass them from the front of the head to the nape of the neck, and then return them. It is obligatory to wipe the hair of the head along with the ears with new water, not with the excess water from the hands. If there is dampness on the palm and one wipes with it, it is sufficient. And if water drips onto one's head, or one places a wet hand on it, or is exposed to rain with the intention [of wiping] and does not physically wipe, it is sufficient.