Adding an OEM receiver hitch
#2
Doesn't need to be oem. Just try to get one of hidden type. It only costs $150 at Uhaul including installation.
As far as the plugs, you should already have a 4 pin wired to your bumper. That'll work for most smaller trailers. If you want to tow larger trailers with brakes, etc, you'll need to get a 7 blade which would have to wired.
#4
No, the bumper would not need to come off. Honestly, just find the best deal, as long as its the hidden style (the bar behind the hitch is either tucked kind of behind the bumper, or may be visible depending on the model), it really doesn't matter and looks exactly the same. If it's a class III hitch, oem or the Curt brand Uhaul uses, it will look no different. You're probably going to just waste money if you get the oem version.