The two most likely things are that you hooked it up wrong or the thermostat is bad. Have you tried putting the old thermostat back to see if it still works? That could at least eliminate any potential problems outside of the thermostat itself.
As far as I can tell most of your troubleshooting isn't really showing much. When you pull the batteries, the thermostat display probably should go dead, as the batteries are what actually powers the thermostat in most cases (not the wires from the furnace). Also, if you're checking voltage across a switch, it makes perfect sense to have voltage across an open switch and no voltage across a closed switch. And those live wires you're playing with are most likely low voltage and not really all that dangerous. But don't put your tongue on them or anything.