How to avoid Copyright issues for news aggregation site/app?

You don't post the entirety of any article. You link users off to the actual site (or iframe it in even or open a new view on a mobile app with a back button). You cite the source. OpenGraph tags (and other meta tags) are typically fair game for previews and such.

Saving articles for offline use is a bit trickier. So long as you aren't storing their copyrighted material on your server you might be ok. There are, after all, tools out there to save offline versions of any web page. Of course they let the user enter a URL and do not lead users to specific pages.

RSS feeds are also typically ok. If you look, many sites have a policy posted about how you can use their RSS feeds. Some sites even put ads in their RSS feeds. I wouldn't alter anyone's RSS feed. They want those ads syndicated. You might have problems there. Might.

Ultimately, you are pushing traffic to them. Hopefully. That's kinda how this works. You want to build something that services them and your audience. That way everyone wins and no one has any problems. So even if they might normally say something about copyright issues, they likely won't because they'd be shooting themselves in the foot.

Basically, make it worth their while. Make sure they don't want to say no.