So one mac mini to two TVs? Would that actually work? I mean wouldn't it be the same content/desktop? I never tried to do it so I don't know.
You may want to consider Apple TVs at the TVs. Use the apps for Sling, Netflix, and Plex app for content from the mac mini.
I haven't used the Plex DVR software, well because I wrote my own DVR software. It's server software that would run on the mac mini and use the HDHomerun (recommend the EXTEND model because it will encode to mpeg4 in hardware) over the network. I also wrote an Apple TV app to control the DVR server software and to watch recordings and live TV.
My DVR software requires a subscription to Schedule's Direct which is $25 a year. I do get free coupons from time to time for people who want to try out the project. Otherwise all the software is free.
I really don't promote or push my project much because I wrote it for my own purposes. And being the fact it's only me it's a lot of work to please lots of different users. But after reading your post I was thinking it could be an option for you since you are just now trying to put a system together.
and calling it j4cc (jflicks for cord cutters).
As for recording Sling TV - there isn't an easy way to do it as far as I know. I mean I don't think there is any project that supports doing it.