Edit: I've just read your second paragraph. You could always build something on your own steam but you'll won't have the same level of integration and convenience as Google Reader unless you spend a very long time on this. Ultimately I think you have to settle for just a web interface and a browser on the desktop.
And out of all of the self-hosted solutions, unless able to build something perfect for your requirement, Fever looks like the best client (albeit at $30). It's all a bit Apple-flavoured but there you go.
If that's too rich, Tiny Tiny RSS might hit the bill. Oh and apparently they have an Android client in beta. You'll still be limited to the web interface on the desktop and it's not the prettiest (neither is Google Reader in all honesty) but worse things have happened at sea.
Edit 2: I've just read your first paragraph (yeah I'm doing things backwards today). If you want proper offline sync, I think it's possible that you want to be running this server on your computer, not an external server. For external access you'd need to expose it to the internet (port forwarding plus something like dyndns or similar). That's the only way I can possibly see you getting something even remotely like this.
I've recently moved from just using Liferea to:
- A Google Reader account (web access)
- Lightread (the new app from the recent app competition)
- The Google Reader app on my phone.
Now I can be made angry by news whereever I am and it's all in sync.