I use ReadYou. Has more features and easier to understand than Feeder (at least for me).
Requires no permissions (except enabling notifications), can use local (on device) accounts. Only requires RSS links, no other data.
Supports notifications, have not tried it. Also likely requires background refresh, which I don’t use, so don’t know about that.
It is definitely possible, as RSS readers like ReadYou can do it. Maybe try FreshRSS in conjuction with an RSS reader?