On the older Taurus's like the 3.8L, the heater core was part of the engine cooling system and can get hot spots in the head if that core is blocked causing even further head gasket problems. My kid has a 2000, I never really checked that out, generally the feed for the core is at that thermostat housing, but it's the return of that core that is of concerned, if it returns to the heads or the engine block, would worry about that.
Would have to go back quite a few years for a heater valve, do have one in my 82 Chevy motorhome, and forgot how difficult it was to set these things to an agreeable temperature due to the delay, blend doors are practically instantaneous and that is what you have.
Replacing the heater core starts with recovering the AC system and removing the entire heater box that contains both the heater core and evaporator, let's just say, it's not a five minute job. Your return hose being cold is definitely telling you that something is wrong, was an electric heater installed, those kind you plug in at night with a prayer your vehicle will start the next morning. Going back in years on this one, many would install such a heater the wrong way that would kill the heater, but a possibility.
Then there is the question of history, did you just buy this car for your daughter or was it in the family for years, and was the heater ever working?
I take heater core leaks very seriously, antifreeze fumes can damage your kidneys, and those are not very easy to replace. This happened years ago to my first cousin, was a distance away, but was driving an old beater, a long expensive painful death that also nearly broke my aunt and uncle.
If my daughter were driving that car, wouldn't worry about an overcoat, but would rather put a bomb in that car and blow it away than to take that risk. Son had a leaker several years back while going to college, money was tight, took me two days to change that sucker, but got it done working late into the night. Not saying this is your problem, but just offering this as a precaution.