EDIT: This only seems to work for audio, thanks for pointing it out

Try the AirCast community addon. The description says:

AirPlay capabilities for your Chromecast players. Apple devices use AirPlay to send audio to other devices, but this is not compatible with Google’s Chromecast. This add-on tries to solve this compatibility gap. It detects Chromecast players in your network and creates virtual AirPlay devices for each of them. It acts as a bridge between the AirPlay client and the real Chromecast player.

Sounds like just the thing you want, although I haven't tried it personally.

Airplay doesn't require Internet access. I don't blame you for not wanting Samsung software phoning home.

Download the movie and cast it?

There’s a TV+ app for chromecast, isn’t there? Or is this one of the older chromecasts that you can only use with your phone?

Have a quick check and see if Apple has made an AppleTV app available for your TV. I have an 2012 LG TV and found an AppleTV app in its store. I can now Airplay direct to the TV.

If not checkout something like the DoCast app in the iOS AppStore. I think it may stream to Chromecast for you.

No.