Graviman,
You are right, 36% is actually good, relative to what is possible. The 250 series has a single centrifugal compressor which looks like the pump in a washing machine or the blower in a vacuum cleaner- a conical shaped wheel that has spiral fins on it, so the air is swept up and spun into an ever decreasing space (since the fins get shorter and shorter as they get to the top.) The wheel spins so quickly that the air does not escape due to its own inertia. The 250 gets a 9:1 pressure increase in this one stage!
widgeon,
The bleed air cooler works just like a refrigerator or air conditioner. The working gas (air for us, freon for the AC) is squeezed and made hotter by that squeezing, then that hot pressurized gas is run through a cooler/heat exchanger while still pressurized. This is really like a car radiator. The fluid's heat is removed (and dumped outside) and then the gas is rapidly expanded so the cooling is appreciable. Often, a turbine (run by bleed air) inside the system is used to increase the pressure of the gas further, so the cooling is more efficient.
Here is a schematic of one such system:
http://www.tpub.com/ase2/73.htm