It seems hard to believe that all of these problems could be related to simple leaks and subsequent air into the system. Would that halt the flow such that fluid is not circulating into the heater core? Why would the hoses to the core be hot to the touch after a run?
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Consider that the car cooling system is the opposite of a home central heating system. If you get air in that the pump finds it hard to circulate the water. Air trapped in the rads can leave them cooler than the others. In your car, air trapped in the heater core will stop water circulating through it giving poor heat and demisting. When bleeding the system the heater valve must be fully open to get rid of the air. It's common for Volvo heater valves to become sticky or sieze up open, closed or anywhere in between. You might have the control set to closed but the valve can still leak some hot water through and so both of the small pipes will feel hot. If the valve closes properly you should feel a difference between them.
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Also.... why would I need a new radiator cap...? I though the thermostat unit was located inside that cast metal unit that flows into the top hose going to the radiator. Isn't that thermostat rather important to the whole operation of things?
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The radiator cap keeps the water under pressure. It effectively raises the temperature at which the water boils which allows the coolant to carry on doing its job before it has to blow off to relieve the pressure. When you have air in the system it expands more than the coolant which forces air and water out at lower temperatures. Rad cap seals get hard or crack making them leak before the marked figure, possibly 7lbs in this case. There are typically two lengths of cap so make sure you have the right one. When a cap isn't working very well it's common for them to blow a couple of minutes after parking after a fast run, as heat soak from the engine boils the water. The thermostat allows the engine to reach working temperature quickly before opening and allowing water from the rad to come in and add some cooling. It then opens and shuts as required to keep the temperature stable. The heater is in the engine circuit so it gets hot water quickly from a cold start.
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Also... how could I diagnose if my water-pump/impeller is failing? I'm also hearing a disturbing clunking noise sometimes, which I've never heard before, and does NOT sound healthy...
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If you slacken off your fan belt you can see if there is any play in the pump by pulling on the fan forwards and sideways. There should none or virtually none. Pump seals can leak as well as the seal ring to the head so check carefully to see which is actually leaking.
"Clunking" isn't a description that's usually applied to a failing water pump. That's usually reserved for the fibre timing gear and that's another story entirely. A quick search here will find lots of information on that.
Stay cool!