Most of these projects use CSS or the DOM to provide animations, which aren't available on react-native. Usually, all compatibility problems between the two come from that.

Not at all. React Native renders native elements, so CSS does not exist, just a CSS-like syntax that maps to native styles on Android and iOS. React Native is finished enough to do mature apps in, as Facebook is already using it in production on their mobile app, ads app, Instagram, etc. Other companies using it include Baidu, Tesla, Wal-Mart, AirBnb: https://facebook.github.io/react-native/showcase.html

It's certainly an evolving product and not every native function (such as video recording) is available as an easy-to-use component. In those cases, you can use a 3rd party NPM component or write your own Java/ObjC and link it in with your own custom component. Even if you have to write a little bit of Java/ObjC, it's a huge time saver.