1. Yes, off-site hosting can be used. When you go to add the item, you can specify an external URL and choose the option to not actually upload anything (so you create a new "empty" download pointing to an external site instead).
2. Yes, given appropriate permissions. You can also specify what types of files can be uploaded.
3. I'm not sure about this one.
4. Yes, there's a setting to require approval of all downloads, and you can give user groups permission to manage the downloads system.