Why does the heater core transfer heat from the coolant, not the exhaust?
 

It would be hot almost instantly and would allow for an air cooled engine whose temperature could be controlled precisely by the amount of air that was let into the engine compartment.

The only reason I can think of is that it would be TOO hot, but that can be solved by just placing less heat exchange material on the pipe. It's probably be just fine to run the regular pipe through the airstream.

Too complicated is my best guess...

If it leaks, it's poisonous gas leaking in the cabin versus engine coolant ;)

That, and it's a whole lot easier to route a relatively clean liquid than a much hotter gas ;)

trebuchet hit the nail on the head....toxic fume leaks.

Why don't they run an absorption cycle A/C from it?

I've been curious of the same thing for a few years now.... But I haven't educated myself enough on the concept to build one myself :p

I have a number of vehicles where the cabin heat is gotten from exhaust heat exchangers and/or from cylinder waste heat (VW, Corvair, general aviation aircraft (but I don't own one now)). That system can be great WHEN EVERYTHING IS IN PERFECT WORKING ORDER. But, reality interrupts our little daydream (doesn't it always?). Over time exhaust systems rot out and yes, noxious fumes come into the cabin. Also, when (not if) there is an oil or gasoline leak in or near any component of this sort of heating system, all that stench comes in the cabin too.

both good answers, I will also add.

-heat is exchanged easier, and more efficiently in a fluid
-antifreeze has an anti corrosive that should ideally keep the system working properly longer
-exhaust is wizzing out of your car as fast as possible, back pressure makes your car perform less goodish, where as the radiator fluid is restricted/slowed down by the therm to make the engine coolant exchange work better, and it doesn't _really_ effect engine performance.

Fire: I disagree with your points 1 and 3. Experience shows those not to be factors.

That's assuming that you and I agree that VW and Porsche heaters are sufficient. We don't.