Does Apple take apps off the store if they crash a lot
I've a couple of simple apps in the Mac App Store. When uploading them, Apple thoroughly test the app, from the functionality of the application itself, to whether or not the app quits when the last window open is closed. The same approval process applies to iOS application loading, though I haven't experienced it personally. 

The applications that aren't working for you now would have worked when they were released. The most usual issue would be an iOS update crippling the app's functionality, for reasons known to the developer.