FYI I just now had a similar issue with my LastPass extension for Firefox. After some mucking around and multiple attempts, I think I managed to get the add-on to update and that seems to have fixed the problem. Might be worth removing/reinstalling the addon to see if that helps.
On the topic of "2FA bypass", it sounds like you've gotten into the "offline access". This is a reasonable security pattern if you think about it (though I can't speak for the implementation). Since you've logged into the device before, it acts as the second factor ("something you have"/computer + "something you know"/password) for offline data, but it will still require your mobile authenticator to resume ongoing online access to your account. Importantly, this won't work for some hacker in a foreign country - they will still need the 2FA code before the addon will download any of your vault data (assuming this is implemented correctly).
From a security standpoint, note that this data is already on the device, and your password is the decryption key, so giving you access is reasonable (it's just user-friendly access to what's already there). Personally I reckon it should give you a clearer option to wipe local data on logout. Nevertheless, in general you should assume that unless you've performed a cryptographically secure wipe then your data is still left on the device. If that concerns you then you should be using disk encryption, BitLocker, etc.
You're right about the cookies. It's weird that an add-on/browser extension would stop working because of this because it has other (more robust) ways it can persist data.
Hope that helps!