Exponent exposes all of the React Native built-in APIs which encompass a lot of what you're asking about.
For example, here is the Clipboard API: https://facebook.github.io/react-native/docs/clipboard.html
Wifi network configuration is more complicated, but there are some things exposed like "NetInfo" which lets you know the current state of connectivity. https://facebook.github.io/react-native/docs/netinfo.html
The complete React Native docs are here: https://facebook.github.io/react-native/ and Exponent specific docs are here: http://docs.getexponent.com/
For now, Exponent just lets you write things that only use JS, and we make that easy and reliable and straightforward. We actually think this is the best way to write most apps that can be built this way since the code is easier to maintain and works cross platform, and you can do things like instant updates much more reliably when all your code is JS.
We're looking at a way you can write Java/Swift/Obj-C extensions for Exponent, but that will take us some time.
With Exponent you write just JavaScript. In addition to most of the APIs that come with React Native, Exponent provides several APIs (https://docs.getexponent.com/versions/latest/sdk/index.html) and we're steadily adding more. If there's a native API we don't have yet that developers need, they tell us about it and we add it to our roadmap.
Exponent is open-source, too, so if you needed to you could fork it and write your own native code or make a raw React Native app. We recommend writing pure JS apps if you can, though. At Exponent we write native code so other people don't have to.