And speaking of a weight distribution hitch. Is that something I can install myself; or is that something the dealer should do.
You can DIY - the hitches come with complete instructions and it's straight-forward wrenching. That presumes your truck is already equipped with an adequate hitch receiver. If you also need to install a receiver, it can get a bit more complicated because it has to be solidly attached to the vehicle frame. Even then, it's still a handyman sort of task. F150's are typically equipped with a receiver, but make sure it's rated for the load to be towed. A Class 3 receiver can handle up to about 8000 lb trailer and 800 lbs of tongue weight. It will have a 2" square tube to accept the trailer ball mount.
Does your F150 have wiring for a standard 7-pin trailer electrical connector? That connects up the stop/tail/turn signals, the trailer brakes, and provides auxiliary 12v power to the trailer when under tow. If not, that's another potential DIY chore.
As Kirk said, you may not need WD for a relatively light trailer on a pick-up truck. But it doesn't hurt either. An F150 ought be able to handle the roughly 400-500 of tongue weight of the trailer you cited. However, I think you will find you want something a bit larger (and thus heavier)..