Sorry for awkwardly entering/reviving the thread, guess I wasn't on this list until now!
Jason, thanks for jumping in! Great to "see" you again :-).
Tong, he's correct that our app is open source and you're free to use anything there that's useful. Our current networking stack is built on top of AFNetworking, and you can find the API calls we're making inside the various "Fetcher" classes (in the Networking folder https://github.com/wikimedia/wikipedia-ios/tree/master/Wikipedia/Networking).
We hope to eventually release a standalone MediaWiki client in Objective-C, or Swift, or both! However, the dynamic nature of the MediaWiki API's requests and responses will probably require us to keep the framework limited to a set of components that allow iOS developers to build requests and serialize responses into model objects of their choice. We can provide high-level abstractions for the most common queries (i.e. the ones we're already using in the app).
We're in the midst of a more UI-focused update at the moment, but hopefully our next big infrastructure push can result in the release of such a library, probably on CocoaPods or Carthage.
In the future, you can email mobile-l with any questions related to the iOS client, and be sure to subscribe there for future updates on the app.
Cheers!
Brian