So when I'm not at home, I want to be able to send links (via my Android phone for example) which will automatically start downloading? And will be ready to watch on my Shield when I get home. Not unlike RSS feed functionality but with the ability to send obviously. Thanks for any suggestions!
I assume you are talking about links to video files..
I use various methods but depending on the source of the video they require a computer or seedbox. I don't think AndroidTV can do this on it's own.
If it's a link to a file on Mega then you could add the file to your mega account then have a computer at home sync the mega folder to a download folder, so it's ready when you get home.
Personally I use a Seedbox for downloading my torrents and I also have SabNZBD on my seedbox for grabbing stuff from Usenet. I can trigger the downloads from my phone and either download to my home PC when I get home or stream directly from the FTP server on the seedbox using Kodi (I've got the FTP server added as a source)
Just use Plex
You can do that via web version of playstore. I don't know if there's workaround for nonplaystore apps.