This is true, but doesn't fit as a solution to the problem stated as

whenever I hit "vote" or "I've checked all these" buttons

... because these buttons should submit to the same domain the original page was served from. Vote buttons or the "I've checked all these" shouldn't be visible for domains where you are not logged in.

I don't have any experience with Opera so I don't know if deleting a cookie once blocks it for all time like with Firefox. I think that Perlmonks sends a cookie to whatever domain you've used to log in - maybe there is some tool for Opera to inspect what headers get received from the server when logging in and whether Opera actually considers the cookie to be valid for the current page...

Update: The easiest explanation would be a wonky localtime setting on the machine. If the cookie is sent with a timeout that is (according to the machine) in the past, the browser would immediately discard the cookie.