I would not have done a compression test unless I noticed a running problem or the customer complained of one.
run compression check on it
use a compression tester.
compression and rarefraction
Remove all the spark plugs and turn the engine over by hand. Place your thumb over the #1 plug hole and you will feel the pressure on the compression stroke. A compression gage will also tell you when you are on the compression stroke.
The purpose of a thermostat on a running heat pump is to be able to determine the temperature the pump is operating at. This is used for regulating the heats needed.
Try to determine the origination of leak i.e. radiator, hose, thermostat housing etc. Try to determine the origination of leak i.e. radiator, hose, thermostat housing etc.
You can have a compression test done on the engine to determine the condition of the rings and valves.
A compression test will determine which cylinder(s) are at fault.
The engine will be losing compression, and possibly burning oil. Check each cylinder to determine the compression. The loss could also be caused by a valve problem.
If you need a heater core, the one you currently have would be leaking. If the car is over heating or not getting warm enough, think thermostat.
Have vehicle scanned to determine problem
When the car is hot, squeeze the upper radiator hose. If it feels soft, then the thermostat is not opening and therefore bad. If you squeeze the hose and its hard to squeeze, then it is pressurized and the thermostat is open. That is one way to check if the thermostat is stuck closed. If the thermostat is stuck open, then in cold conditions the car will take a long time to produce hot air from the heater.