Does a voice coil vibrate at the same frequency as the source sound file?
I would expect so, assuming the speaker is capable of vibrating with the same speed. 

As far as I know each speaker has a range in which it can vibrate (usefully). Mostly bigger speakers are better suited for vibrating at lower speeds (and having more 'pressure', more air that is moved), small speakers can resonate faster (thus higher frequencies).

That is why for lower frequencies you need bigger speakers, and for higher frequencies smaller speakers will do.