2002 fx4 5.4
#2
Look at your payload sticker on your door. I would say that you probably would be able to, but check you sticker. The payload part is what is going to get you with that scenario because your are putting a lot of load "on" the truck with the golf cart. That trailer is hardly nothing as far as payload since most of the weight is on the trailer axle and little through the tongue.
#3
You should be fine with it in the bed if you have the length. I wouldn't trust the tailgate cables if its a short box. Put a couple boards under the rear tires if thats the case to carry some of the weight. I put my Rhino in the bed of my '98 many times, sits level-just below level and its 1200lbs dry, larger wheels, fluids, winch, few other things its probly more like 1400lbs.
#4
Grumpy Old Man
The properly loaded trailer should have a tongue weight less than about 400 pounds.
Load the pickup up with everything that will be in it when towing. Driver, passengers, pets, tools, jacks, and the golf cart. Go to a truckstop that has a certified automated truck (CAT) scale and fill up with gas. Then weigh the wet and loaded pickup. Subtract the weight of the wet and loaded F-150 from the GVWR of the F-150. If the answer is 400 pounds or more, then you'll be fine. If the answer is less than 400 pounds, you'll probably be overloaded when on the road with the trailer tied on.
Or if you already have the trailer ready to go camping, then weigh the wet and loaded F-150 with the wet and loaded trailer tied on. Add the weight on the front and rear axles of the F-150, and compare to the GVWR of the F-150. Then you'll know where you stand.
Any answer that doesn't involve weighing the wet and loaded truck is just a WAG (wild hiney guess).