It doesn't matter whether the van is parked and everything is turned off. The separator operates independently. It engages when the voltage on either main terminal rises above 13.4, and disconnects when the voltage drops below 12.8. The specification for these trigger voltages is plus or minus 2%, so it could stay engaged a bit below 12.8. The way to test the separator is to turn on a significant load like the headlights (without the engine running or the shore power connected) which will pull the voltage down well under 12.8, and the separator should disconnect. It should reconnect after you start the engine or plug in the shore power.
The solenoid coil in the separator draws 0.75 amps when engaged, which means it dissipates about 10 watts, so yes, it gets pretty warm. Too hot to leave your finger on very long, as I recall. As discussed in other threads in this forum, this separator is a pretty crappy unit and should be replaced by a Blue Sea relay if you suspect it is malfunctioning.