Newbie / Towing

Hello all. I have a new 2012 F-150 Supercrew 4x4 with the 5.0L and 3.55 axle. I own a 2008 bumper pull travel trailer that has a GVWR of 10770 lbs. I am thinking of trading in the travel trailer for either a lighter bumper pull or a 5th wheel. I'm new to the whole travel trailer world as I purchased the one I have now used last year. I have tried to research my new truck's capabilities to determine what it can and can not haul/pull safely and seemed to have only confused myself even more. Ford's website and my owner's manual both say my truck has the following capabilities;
7700 lbs maximum towing
7350 lbs GVWR
1570 lbs maximum payload
13500 lbs maximum loaded 5th wheel trailer weight

So does this mean I can pull a 5th wheel up to 13500 lbs as long as the pin weight doesn't exceed 1570 lbs?

Can someone translate all this please?