Looking back through this thread I think it comes down to this. You are not entering the correct Gmail password that matches up with the Gmail address. Because you had 2FA enabled on Gmail, the password entered in Outlook 2010 was an "app password," not the regular Gmail password. That app password will only work for the Outlook 2010 app, and not for logging into Gmail using a browser. Why the login of the Outlook 2010 email app failed I don't know, but it may have to do with Google making some changes that disallow insecure apps from being connected to Google accounts.
In any case, you should be able to reset the main password for Gmail by the automated process if you are supplying the correct information for the account: the backup email address had to have been set previously, as did the phone number for the code. If those things are correct, be sure to use a computer that previously logged into Gmail from the same location--Google checks for that with password reset attempts.
Going back to Outlook 2010, you might want to read this about app passwords:
https://support.google.com/accounts/answer/185833?hl=en