No, you can't have two thermostats wired to the same furnace. The thermostat in the hallway is a heating and cooling thermostat and it is almost certainly wired to the "furnace" in the attic. The evaporator coil is for the air conditioning and is attached to the furnace in the attic and uses the furnace fan to circulate the air.
Furnaces don't have water pipes, they have 3/4" gas or oil pipes and they have ~3" combustion exhaust ducts that are sometimes made of pvc piping material. The word "furnace" referring a device that burns fuel to heat air. If you have a boiler, and you really have two water pipes going into the air handling unit (are they insulated?), one would be the supply and the other the return.
Now it's clear what furnace really does. I don't know about the air handling unit. The way you describe sounds like the hot water gives off heat to the air handling unit for air ducts. The ducts seem not insulated. Hot water copper pipes are not insulated,too.
Right, 62F is the heating setpoint -- but it looks to me like it is also the cooling setpoint. I can't tell from the pictures if the thermostat automatically changes between heating and cooling. Between "heat" and "cool" is there an "auto" that it is set for? Or just "off"?
I think it is. I can see in your first picture, the pointer in the upper left has options of "heat" and "cool". I think the unit in the attic has heating and cooling because the thermostat indicates it is for a unit that has both heating and cooling.
I see but I still don't know why I don't get heated air. I will ask the maintenance man.
By leaving it set in "auto" it will change between heating and cooling on its own. Or just accidentally turning it on and not realizing that it is both a heating and cooling thermostat.
It isn't difficult to tell what a system is doing: if air is coming out of the supply vents, it is on. If the air is warm, it is heating and if the air is cold, it is cooling. And when it is cooling, the condenser outside will be on/make noise.
It might be worth having a friend who knows about such things - or even getting a contractor - to come and poke around, looking at your equipment to tell you what you have. It isn't easy to tell from the limited and confused information you are able to tell us.