Does water constantly flow through the heater core or is there a valve that shuts it off?

Generally speaking, there is always a valve. Otherwise ac ertain amount of heat would always be radiating into the cabin. Depending on the type of climate controls you have determines what type of valve you have. I have an H1 Hummer and had to install supplemental ball valves to shut off the water flow completely because the cab would get so hot because the A/C was marginal.

Does water constantly flow through the heater core or is there a valve that shuts it off?

NO, it always has water flowing in it.
As for the ac, there is a trap door in the duct work that directs air to or away from the heater core.
Dan

Does water constantly flow through the heater core or is there a valve that shuts it off?

dwrat is correct, the EGR/Heater core loop is a bypass loop that has constant flow.