So I don't know anything about the math or physics of the situation, but I do actually work on cars. A car has both and evaporator and heater core. Meaning it is capable of using the evaporator core to act as a dehumidifier before the fan from the blower motor pushes it through the heater core and up into the cabin. The heater core gets it's heat from the engine coolant. When the heat is activated a little internal door opens and runs the coolant through it. I'm assuming you'd understand the rest. But thats why it takes time for your hot air to blow. The engine hasn't heated up enough to transfer it to the coolant.

That's likely because you also happen to feed your cat. The statistic about birds killed by cats is referring to feral cats that hunt for survival being the bulk of the problem as far as I'm aware.