Not sure anyone can understand the problem, too many details that sound unrelated from what I can make out.
Let's just use this part of your post:
On the website.com that you see the folders, are they from that same account that works at
www.website.com or no? Does this account have an index.html or index.php in it's public_html directory?
You might try renaming the .htaccess file in the root of that account to .htaccess.bak for a moment and try the 2 URLs again just to rule out something wrong there. Be sure to rename it back to normal after testing the domain.
If I'm completely off base here you might try explaining it a bit more clearly so someone
much smarter than I can help you out.