If you applied heat to two identical pots, one with a quart of water
and the other with half a quart of water, then at a time estimated to
be half way before the quart begins to boil, you added the other half
to the half....which one would come to a boil first? Or in other
words, should I warm my car up with the heater on, or wait til the
engine warms up and then turn it on/

Unless your car's heating has an electrical override that detects when
cold air is being pushed through the system, you'll just be pumping
cold air into the car.

On the other hand, having the extra load of the fan running should put
a little more strain on the idling engine, which would probably make
it generate more heat.

If the car is a real classic, for example an Aston Martin, then there
are other considerations. An engineer who owned one told me that it
was essential to let the engine run for five minutes to get the oil
warmed up and circulating freely.
Adding any extra load in that case would be ... defeating the object.

If your vehicle is modern, then it probably does not matter much, the
bearings and oil pump(s) will be much more robust.

Still, pumping cold air into the cab seems a bit pointless ...

OTOH if you are nipping out to start the car, then retreating to the
warmth while it sorts itself out, then there is the personal comfort
factor.

If you want it in terms of the original question. Suppose at the start
pot A has 500 mls of water and pot B 1000, both at 20oC. When pot A is
half way to boiling, it will now be at 60oC (an increase of 60oC): pot
B will have only risen 20oC, and will be at 40oC. If you transfer half
of pot B to pot A, then pot A now contains 1000 mls at 50oC while pot
B has 500 mls at 40oC To bring both to boiling will take 1000x50
calories (50000) for pot A but only 500x60 (30000) for pot B which
will boil first

That's a LOT of questions but the simple answer is that it depends on the car.

The only way to really know is to conduct a simple test for yourself.
That's what a scientist would do.

Start the engine and the heater at the same time.

The argument of "you'll just be pumping cold air into the car" is
invalid.  Isn't the inside of the car assumed to be the same
temperature as the outside air???  So by pumping in the outside air,
you won't be decreasing the temperature inside the car.

The benefit of starting the heater at t=0 is you will be gaining heat
AS THE HEATER WARMS UP.  If you were just to wait till the heater
reached its max output to turn it on, you would have missed the
opportunity of gaining the 'ascending' heat.  Collectively, this is a
significant amount of energy.

As for the wear on a cold engine, I wouldn't worry about it.  The
heater is simply a resistor and a fan.  Not much load...  It's not
like starting your car and then zooming off at 100 mph.  The idling
engine's power is much greater than what is required by the heater.

Actually, I envisaged that the car was in a cold climate, possibly
below zero degrees centigrade.

I had pictured someone in Minnesota entering their garage, opening the
garage door, starting the car, and scuttling back into the house for a
final coffee.

I should have made those assumptions clear.

In a mild climate the heater would have little effect, as it would
probably not do anything.

Just a suggestion - in cold climantes at home do what I do, just plug
in a small 1200 watt eletctic heater on a timer to start every morning
at a reasonable time.

Or, as I do in my equipment on the ranch, get one of the $10 electric
heaters which run off the battery and turn it on when you start the
vehicle - instant heat.