I'm the author of a couple of modules that allow your RN app to operate in the background:
https://github.com/transistorsoft/react-native-background-fe...
https://github.com/transistorsoft/react-native-background-ge...
Both of these modules were ported from Cordova and have since been ported to NativeScript.
Stuff in the background would be like video/audio calls, playing music, actively monitoring motion/gps (MapMyRun/Ride/Swim/etc), and a few other things that escape me at the moment.
If you are looking to release games quickly, the choice is generally Unity, although I hate how all the Unity games take forever to load, but it's a great way to get your game out quickly, and the good games will overcome the load times issue.