The supply to the heater core is from the water pump before the thermostat. The coolant out of the heater core returns to the cylinder head, so I'm pretty sure the thermostat doesn't affect the coolant flow to the heater core.

 

Yes it is before the thermo but if the thermo is stuck open it does not matter where the supply is your coolant does not get to the vehicles designed temp. Meaning your supply will be cooler than it should be. Am I wrong in thinking that

 

In the first post, he states the car runs at normal temperature, so the coolant is getting warm. The only explanation is that the heater core is clogged, or that valve is shut even with movement in the arm/cable.

 

Yes that is correct. For some reason I skipped over that. If the car if running at operating temp it could only be the valve or core being blocked as you said.

OP if you are sure you your valve is working correctly (suggest removing and making sure it is operating correctly) I would flush the core to clear it.

 

Sounds like a blockage somewhere.

I haven't taken out a valve on a 2007. It looks quite different than the 1998 so I don't know if it's plausible for the lever to be working but the internal valve is stuck closed.

Have you done any work? Opened the thermostat housing?

You say the coolant is full. Have you looked in the radiator or just in the overflow bottle? A bad radiator cap can cause the radiator to go empty while the overflow bottle is full.

Disconnecting the hoses somewhere sounds like the next step because something is blocking flow through the heater core.
- Valve broken internally while the cable & lever still "look" OK?
- Some kind of goo in the cooling system plugging up the heater core?
- I think it's less likely to be the thermostat because I presume the upper radiator hose is getting hot like it should.

 

No work has been done on the cooling system. It's not overheating, temp is steady 1/2 way between H and C. Coolant was topped off through radiator cap. Radiator had no visible gunk, cap was clean. I'm going to disconnect hoses going into firewall and flush. I'm assuming, if there's no clog, water should go in one and come out the other (?). I'm still wondering why both hoses going through firewall are cold. If there was a blockage, wouldn't one of the hoses still be hot?

 

If the heater core is completely clogged, then you'd have no flow through the hoses. The coolant could eventually heat up, but it would be slow. It sounds like you have a blockage in a hose or the heater core.

When cleaning, you should get flow out the other hose. When flushing, you should put water into the outlet of the heater core to reverse flow. The outlet of the heater core runs to a clamp on the thermostat housing and back to the intake or cylinder head.