My 08 Edge has a factory installed hitch, so I can't foresee any problems to put one on your MKX. The factory hitch is a light duty 1.25 inch square receiver, not the std heavy duty 2 inch square. You could probably do it with factory parts, check some junkyards.
If you are handy with tools, you should be able to find a hitch on the internet from Reese or another provider that you can install youself. For the lights its easiest to take the vehicle to a U-Haul Hitch installation center and let them do it as they will have it done in the time it takes for you to read the wiring directions.
I have never looked at the factory wiring. For many vehicles you can buy a trailer wiring adapter. It usually plugs into the wiring harness at the rear and provides a tee that has the wires for the trailer plug. No idea if one available for the Edge or not.
To be honest I have not used the Edge for any towing. I have full size trucks when i want to tow a trailer.