Try putting the paper clip in #85 and turning the key on but put the other end of the testlight to power at the battery. If the light comes on with the engine running and the AC on the PCM is trying to turn the fan on.
So far you have verified that you have a good fan, fuse for the fan and wiring to the fan. Now we start to work on the system that tells the fan to turn on.
You said your air control does not work? The AC button does light up?
Try this after testing at pin #85, put the relay inplace and disconnect the coolant temp connector from the sensor and use the paperclip to jump acrosss the pins of the connector on the harness side.
The CTS is located here
This is on the drivers end of the engine just below the fuel pressure regulator and above the upper radiator hose inlet fitting to the engine.
Turn the engine on and the fan should run, if it does not swap relays with one of the others in the same relay box and try again.
Let me know what you find.
Sunday is "dinner" with my 93 YO mom so I will be offline for a couple of hours.
Let me know what you find and I will post back or I will opt out and allow other experts to continue with your question if that is what you would like