Here's a short answer to all of your points:
- You don't need to give them access via FTP, there are tons of scripts to handle uploading and browsing files on a server, together with permissions. elFinder is a very good script
- There are a lot of articles on user authentication. PHP has one here, and you can find already made scripts to use or reference, like uLogin and UserPie
- You could use the jQuery-based jPlayer. It can be used as a media player / media library for videos and audio. Here's an example from their page that includes playlist functionality.
- You can use elFinder as a file browser, as mentioned above
- Since they will have a common place where they'll be uploading files, you can just configure jPlayer to switch to each user's folder
- All of the above are based on web technologies, so they all should work on modern mobile devices. You will need to follow a responsive web design approach.