I have a bit of experience towing. I towed my open trailer and car (just under 5000lbs) with an Astro AWD and Trialblazer. Encountered trans failure in both veicheles which were rated to tow 5000lbs. I think it has something to do with towing at or near max towing rate. I would not tow a car with the Explorer.
I would just get a hidden hitch for the Explorer matching the rate of what you plan to pull. I don't think your going over 2000lbs. A trans cooler isn't necessary but might make you sleep a bit better.
I picked up a '01 Mustang GT convert with an auto. I got a hidded hitch for that model and it fits perfect. Hitch receiver has a S-bend which clears the rear bumper cover. Car has a 1000lb factroy limit which is good for a wave runner and the other small trailer we have. Mustang will tow the stuff under 10 miles to the beach and back. I didn't add anything to the stang.
I was able to find a plug in (no cutting and splicing) wire harness for the mustang. The y-connection plugs into the left and right factory harness. I leave it rolled up in the trunk and then just run it out of the trunk to the trailer and shut the trunk lid.
Link to the hitches:
http://www.reese-hitches.com/
My wife has at least as much expericnce towing as myself. She takes the 2500 Silverado crew cab with her 30' 5th wheel trailer by herself to go camping. Should say used to. Trailer was sold this spring.