Your questions are spot on. The one point I think you may be missing is that the aircraft can be on the ground with the engines off and still have an ACARS handshake, as long as the aircraft main battery is on.
So, the aircraft could have landed after, say, 2 hours and still have an ACARS handshake after 8 hours. As long as the aircraft battery is on. Also, the battery can be charged by the apu (auxiliary power unit).