Arek, I agree with your comments.
What troubles me is your comment regarding, "It is not hard to squeeze the upper rad hose, but I figure this is normal if the fans are working all the time - the pressure in the system does not go up."
If the cooling system is full (no air) and the engine is hot, the hoses should be fairly hard (all that non-compressability of fluids and all). So.....the fact that they are easy to squeeze leads me to believe you have air in the system.
Thus, the heater issues.
Also, if the fans see hot coolant, they will run trying to cool it off.