I checked the site loginless.com and the file you sent me (proxy.php), and here is what I found.
I cannot see by this php code why the site doesn't access to services like rapidshare or FB, as the file you sent me is responsible for parsing the content, and not for login.
That is why I concentrated on the second problem (with video not played on your site).
I saw that some links (including players) refer to relative paths to video files - and of course there is no such files (files by this address) on your host. That is why it is not played. This problem can be solved by setting the full paths instead of relative ones. I also think that the problem with downloading videos and accessing the sites like rapidshare - here they use AJAX queries to relative paths, but this problem cannot be solved by changing the paths to full ones, as cross-domain queries are forbidden.
So I am not sure how to fix that - we can try iframes or if cross-domain queries are allowed, we can use it. Need to check with specific sites, how all is arranged. And also see the login script.

If I am right for the video playback problem, this is fast to fix, if not - we need to investigate more to find the reason and fix it.