RSS

I use it. I use newsboat with an every-10-minute cron job that checks it in quiet mode and throws up a notification if something is found. A lot of sites don't offer RSS anymore, but you can use snownews filters to fake it. When I want to watch slow 8ch threads, I use a filter that converts a thread to an RSS feed, with posts as articles. For sites without a good JSON API, something like BeautifulSoup tends to work well enough to get what you need done.

I've wanted the same thing; a news reader that will automatically fetch my files (mostly so I can pull pannenkoek's videos right when they appear). I haven't found anything that facilitates this. Newsboat isn't very flexible. It has podcast downloading support, but that's incredibly limited.

With newsboat, you can set the "browser" option to be a script that will do those things, as that command gets passed the url. You can further set up the script to be pluggable, defaulting to your local browser, using your mailcap, or tapping into your xdg-mime settings. Granted, this won't help for feed items as they're fetched by newsboat, only as they're opened for reading, but it's better than nothing.

Use elfeed, a package for emacs.
You will want to just use one of the hooks elfeed exposes to run your dumper.
This one would be tricky to do. Would be easy if instead you wanted to keep 100 unread manga.

Write a shell script and set it up to run automatically. I made one to keep my youtube archive up to date in a few minutes, no need to install a "package" for an already bloated text editor.
Jewtube is easy to get videos from via RSS because the video ID has its own unique line that can easily be obtained via something like grep + sed and passed to youtube-dl.