The LCD on my car radio doesn't display all the digits. It appears to work better if the temperature of the cabin is cold. Gradually, all the characters disappear. If I press on the display, some (but not all) digits return.

The ambient outside temperature sensor is normally located in the front grille area (outside the cabin) and is only there to display the outside temperature inside in the display.

What you call the 'Solar Sensor' on the top dashboard fascia is an ambient cabin temperature sensor, which does relay cabin temperatures to the BCM

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This contradicts and looks nothing like it

https://shop.advanceautoparts.com/find/2004-jaguar-x-type-cabin-air-temperature-sensor.c36050472

There also a sensor in the rear of the mirror to sense the ambient light and turn the headlights on when it is dark and you are in 'Auto Mode'. That one is nothing to do with the AC. On my car that is on the upper dash.

When the car is in sun, the bodywork is heated by the sun and your AC is struggling, when it is in the shade, it isn't and is cool.

Take it back. There is either a leak from when they replaced the parts, or there is a new issue. A pro grade scan tool computer will diagnose the issue based on data from the ECM and BCM.

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because the temp difference between the hi and low side will be greater faster than a normally charged system the reason it cycles off is the keep the evap. from freezing up, but running to low on Freon can cause poor lubrication in the compressor causing damage to the compressor, yes it will blow very cold for a short period of time with a low charge

It is likely the integrated circuitry on the switch control but it would have to be diagnosed with electronic test equipment outside the GM shop but a GM shop could likely give you a better idea and diagnosis. There migth also be a recall or service bulletin so it would not hurt to speak to a GM service advisor or shop foreman.

Cold air in the cabin suggests that the coolant level is low. Check that first.

I’m happy to assist further over the phone at https://www.6ya.com/expert/jeremy_d728a59f986299fa

Sounds like on your first attempt to jump start the car the positive and negative cables were reversed blowing your main fuse. Your interior lights are usually controlled by a switch next to your speedometer or the dashboard. If the switch does not bring the lights/gauges back then it is possible that something else shorted and the vehicle would need to be taken in for service.

if lcd display check for power and ground. When they go bad you have to replace whole radio. sometime you can find a shop to put in a new display.

Sounds like a fuse or an earth wire loose in the cabin. Take the fuses out and chk them in the light. Go thru all earth wires both in and outside the cabin, something is loose if not a fuse, remember fuses in the engine bay. cheers

Check for low coolant, you might also have a sticking thermostat. Alot of fords get air pockets, make sure if it needs coolant to let it run for about 10-15 min with the radiator cap off.