Its a little involved to set up, and you may need a little work to do it the way you need it exactly, but ttrss with the right plugins and setup would let you so all these.
I currently run ttrss off a lighttpd/postgres/php-fpm (a lot of guides do it with mysql but php is a much better option) - so you would need a either a home server (or vps or better) or at least hosting of some sort, and you can use af feedmod to convert truncated articles to full ones or embed original (its a built in plugin) to embed the full article in the feed where possible
You can use any one of many ttrss clients for android to download and view rss feeds - I currently use ttrss-Reader fork with linkbubble so I can read full feeds inline. This caches articles and images if set correctly, syncs with the main ttrss server, and generally provides a nice usable interface.
I'm not entirely clear on the saving articles bit - I tend to just send articles to myself over pushbullet, which is a pretty clean, well integrated way of doing it - there's chrome, ff and desktop windows clients that work, and it sends nearly anything. There's also a publish option that works with chrome on android and the client I mentioned earlier that saves it as a seperate rss feed viewable from ttrss.There's also a bookmarklet that allows you to create a custom rss feed of arbitrary links you can use from chrome (enable the bookmarklet plugin, then go to preferences/feeds/bookmarklet). You can then just drag links to the bookmarklet, and review them from ttrss later.
As set up, this is essentially free for me - Hosting is the only thing that would cost money, but I roll my own on spare hardware (and there's some handy features on the android client I use that handles using different server information inside my lan), but there would be some setup time involved. IMO the flexibility and security is worth it. There's nothing worse than losing your RSS feed host (and I had that happen to me twice), and this gives you quite a lot of the features that google reader gave you (and more)