VHS(C) tape speed issues - how does the speed change for different modes?
Just like with audio tape recording, there is a capstan and pinch roller that controls the tape speed during recording and playback.

If the playback is faster than it should be, it means that either the tape is speeding up during playback, or it's slowing down during recording. The latter seems more likely, given your description.

This means that the tape is not being firmly gripped by the the capstan, and the usual causes of this are hardened rubber or a weak spring on the pinch roller.

To answer your side question, the different speeds are controlled electronically, by varying the speed of the capstan itself. During recording, the speed is locked to the frame rate of the incoming video signal (which, in a camcorder, is controlled by a crystal). The capstan motor and the head motor must be properly synchronized (to each other and to the incoming video signal) in order to lay down helical tracks on the tape correctly. During playback, the speed is controlled through a servomechanism that uses the sync track as a reference.