You almost have it right.
First, to use the Chromecast, you first need to be streaming the Slingbox on a mobile device (iPhone, iPad, Android smartphone, or Android tablet) using our Slingplayer app. Then, from there, if your mobile device is on the same wifi network as a Chromecast, you can then hand off the stream from the mobile device to the Chromecast. So, the video signal goes from the cable box, to the Slingbox, then across the internet to your mobile device (once you hand the stream off to your Chromecast, it will then be going to your Chromecast instead of your mobile device). Here is an article to reference as far as the Chromecast is concerned:
Also, the internet speeds that matter as far as picture quality in a Slingbox is concerned is upload speed at the home location, and download speed at the remote location. We recommend 2.0 for both those speeds for a stable connection for the Slingbox. And yes, streaming the Slingbox remotely does use data at both locations.
Hope this helps!