After playing piano I have the urge to stretch my wrists even though I don't move them that much, why is that?

Swelling. The tendons that go to your fingers that move them up and down to play piano run across your wrists and connect to muscles in your forearm. The entire tendon may swell with playing so your wrists may feel tight as well as your fingers. Stretching your fingers, wrists, and forearms before and after playing is a good idea!
Not sure. I am not sure I have an explanation but the held position may be causing you to be "stiff".