Not, it was a “WTH” populair topic. Some thing you can work around, but mainly the saving is done in the browser via a cookie and not in homeassistant.
No, it won’t. Just allow to save the cookies in the respective browser (not only for this session) and all works. I’d suggest to setup a different user for the tablet.
This is a weak answer. Cookies aren’t the solution for user preferences especially if the user can login from multiple devices. A preference for Dark Mode shouldn’t be stored in a cookie.