I've just experienced this and can tell you how I managed to fix the problem and why it happened :)
For me (at least) it was to do with the domain name. I entered the site and logged in using www.perlmonks.org but all links point back to just perlmonks.org (without the www.) and so the cookie isn't recognized and you get logged out again.
If you login twice via www.perlmonks.org and just perlmonks.org it should fix your problem :)
Edit: Whoops I missed your follow up - anyway it was kinda similar to my experience so I'm glad you found the solution!