Viagra effect......give it a few hours to go down....
Sounds like a timing issue. Could either be the motor or electrical. I've never changed a windshield wiper motor/motors so I'm not sure just how involved the project would be. I do however recommend looking the problem up on You Tube. I once had to fix the driver side window on my car and a couple of videos showed me everything I needed to know about getting the interior panel off, window out and reinstalling everything. Saved me a bundle.
No, they should go back down. Your switch may be faulty. Or else you are turning off the ignition before they go all the way down.
True, if you turn off the key before they finish their cycle they will stay up.