That is pretty weird all right.
I can't think of anything in Moodle that would cause such inconsistent url patterns. But here are a few ideas off the top of my head.
Filters that change text on the page can mess things up, if they filter and try to replace common words, but I doubt that would affect things like the admin menu. Anyway to test that you would turn off all the filters from the site admin -> plugins -> filters -> manage filters page and see if it fixed things.
Also you could have dodgey theme customizations. Try changing the theme to a standard one, and remove any additional html from the Moodle menu. (site admin -> appearance -> additional html)
Possibly there is apache rewrite url mischief going on? Check for an .htaccess file in your Moodle root. More info on that is here. http://httpd.apache.org/docs/2.0/misc/rewriteguide.html
If the domain changes, as it sometimes does when you upgrade or move servers, admins commonly do a search and replace on the whole database to update any static links stored in text areas or places that don't use urls from config.php. If that was done poorly you might get problems. But again that should not affect the admin menu links. Anyway, info on the search and replace is here: http://docs.moodle.org/22/en/Search_and_replace_tool
I hope something in there helps.
Justin