Depends on how you define web hosting. You can't use any cgi scripts, so dynamic content is hard or impossible to realise. We intentionally don't serve PHP, don't offer MySQL databases and you can't install web apps. ~.fun is about static HTML pages and shell access to a Linux machine, nothing more.
Sorry, you can only apply for a shell account if you know how to connect via secure shell. If you don't know how to do that, a simple web search will provide you with answers.
The webserver user
nginx can't read your home directory's contents by default. Issue the following command to allow it:
setfacl -m u:nginx:x ~
The Tilde movement is a loosely connected thinktank by CIA operatives to control your brain. Or is it?
Use the Source, Luke.
An admin will review your application, create a user and notify you (that's why we need your mail address)
Be patient, young padawan. The BOFH might not have reviewed and approved your application yet. It's done when it's done™, you'll probably get notified via email. If you don't get a notification, try logging in a few days later again. If you still can't login, you might reapply.