always always always try incognito mode and log in and see if it still does it. If it doesn't it's something borked on your end and you need to just refresh the site by clearing data and logging back in.
different cookies. I'm not sure why your device doesn't work and it's not a 3rd party cookie. It's an https cookie and that's how it works. Maybe try clearing everything and trying again including logging out. if you use incognito mode you'll always get it on a new session. Sessions are also sticky in our setup so if you have gone to one server you will stay with that server via cookie as well.
again, there's a literal technical implementation of how it works. That's not how it works. It works with a cookie, if you have the cookie you skip the challenge and go to the site. If you are seeing it all the time you are clearing the cookie or it's expiring prematurely, but it's not a server side (my) problem. It's a you problem. Just like the fucking embeds are not actually my problem, but I'm fixing it anyway because I hate twitter embeds and how they work.