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That is a very unusual set of symptoms. I need the brand and model number of the outside unit, and I need to know if there is also a furnace or air handler inside the house, and the brand and model number of that.
Please attach a well focused, close up photo of the wiring diagram glued inside the service panel of the unit inside,you can use the paper clip icon at the right end of the toolbar at the top of this dialog box, Include a picture of the thermostat wire connections to the printed circuit board in the same area.
If there is no paper clip icon, look for the '+attach files' link at left of your send button.
I can mark it up with trouble shooting instructions. You will need a multi meter ($30 at a hardware store) and some experience working with appliances, let me know a lot about your skill sets in those regards.
If none of this in your skill sets, tell me which direction you are from the center of the nearest city and I will look for a few well rated service outfits in your area.
We can go from there.
Look that over please, answer my questions regarding the piping, and what you mean by the 'pump'