No, you don't need to download Drive for the computer if all you want to do is edit documents online. Drive for the computer is meant to sync all of your non-Google Docs related files to you Drive account, and it also helps to allow you to edit files on your computer while you're offline (offline editing)
As far as the one account that has lost all of its documents is concerned, can you provide more context? Is the user signed in? Do you have access to the associated Gmail account? What are you seeing when you try to view your docs?