the rod has a ball up top and goes down from the handle and into the lock mechanism, where it is held by a clip. the clip that holds the groove in the rod "can" be slid down to release the rod but it is a real pain (at least for me, i just finished paint and reassembly).
the rod will not drop off the handle, there is a clip in the button, and the slot doesn't seem to allow it to drop out anyway.
you do not have to pull the lock cylinder, i removed and installed one of the latches 2x without re-pulling the cylinder, although its pretty easy to pull the clip if needed.