Chromecast does not cast content from your device, your device tells Chromecast where to grab the content from the internet and then just uses the device for playback controls (ff, rw, pause, etc.) It does this to get the best resolution for the target playback device, not your phone or tablet.

The only thing you can really do without the internet is casting from your chrome browser via the cast option. I believe that allows you to cast the chrome browser tab contents directly to your Chromecast device without going over the internet.

The Chromecast is designed to work only with internet. Some people have had luck with creating a hotspot with their smartphone. It didn't work for me tho.

I'm pretty sure the receiver app that loads during a cast needs to be grabbed from the internet.

You don't, you can screen mirror your phone though