Did you enable the cronjobs? The link checking is done with a script (which is needs direct access to the internet, unless you do it with a curl enabled version -- see pubsvn.ez.no)
I would not use another link checker, for us the ez link checker works fine mostly.
If I click Open in new window, I get "the object is not available".
If I click the Url listed, I get: "URL Information" - This url is no longer available.
Are there any documentation on this? To check if things are configured the right way and so on`?
Yes there is a "bug", and I think I did not report it yet: when you have newer versions published, the linkchecker ignores this and also looks at the older ones. Also objects that are in the trash remain the target for link checking. The linkchecker should skip these.
So try clearing the trash and see if this helps (as a workaround for now)