Yes, Apple provides Siri (voice search, automation, calculator, and much more) and music control plus some notifications that respond to touch when on the lock screen.
No, in that you can't choose any app to actually run from the home screen before a passcode or slide to unlock action.
Back to the yes aspect of the answer, touch and Siri's access is very strictly limited and you don't get to use any apps in their full UI, but instead use Apple controls to automate things like changing volume, skipping to the next song, pausing and playing.
Similarly, when you use Siri to access things, you can generally input a new appointment and interact with data stored in an app as opposed to firing up the app's UI (in the case of calculator or other apps you ask about - like perhaps choosing a new playlist in pandora). Siri of course knows how to play music in the library, but not yet how to select a pandora playlist.
The prevailing idiom is to select specific apps that themselves have passcodes (banking apps, finance apps, secure work apps for medical data, etc…) and not worry about a device passcode or to just enter the device passcode to run any and all apps.