Some newer vehicles require a special machine to purge all the air out of the cooling system. Air trapped in the heater core would do what you described.
If you put detergent motor oil (like what you'd put in your car) in your air compressor, it will turn milky looking. You want to run a non detergent oil. For home use, it doesn't matter if it is conventional or synthetic.
What's your budget? Some very reliable units are still made for a high price. A used industrial compressor can be had for cheap if you can wait for the right deal.