One quick thing to check that I don't see mentioned...
It sounds like there's some other things going on, but just in case. Check your antifreeze in the low pressure auxiliary tank, it's the smaller of the two located at the front driver's side of your engine bay.
Mine had a slow leak, and the only way I found it was because when the level got too low, the compressor started rapidly short cycling in the same way you describe. Turns out it does that to protect the system and keep it from overheating. Refilled it, and runs fine. Cap is leaking pressure and vapor and will be replaced soon. I don't believe my scanner showed any trouble codes, but to be honest I'm not entirely sure I thought to check them until after the problem was fixed. Mine also would work perfectly when first starting the car cold, but once the system warmed up it would short cycle the rest of the drive.
Quick, cheap, and worth a look just in case.