Outlook won't let me sign in to my account neither update the 2-step verification information.

I'm seeing many people having similar issues with Outlook, it's due to Outlook and Windows updates (usually) and a mix of Personal Outlook accounts having a security problem. One can cause the other. For you being locked out of the recovery account, that can't be fixed unless you put in a recovery account request with MS through their portal. For the account you are having security issues with, have you tried logging in online in a variety of browsers and also incognito? Try the top ones: Edge, Chrome and now Brave seems to be a favorite. It might allow you to chose another way to verify you. Guessing they are pushing security for Outlook due to the reason I mentioned in the beginning. If you can get in online, you can change the recovery account or remove it and add on 2-step with your phone.