1)  Typically, this is the size of the pad, but your truck can often fit beside or near your trailer.  When planning a trip to a national park, I prefer to not reserve blindly online and call in to make sure my truck & trailer will fit in the desired space. Talking to a real person can save headaches when showing up at a national park!
2) I like the way my truck pulls with a weight distribution hitch and sway bar.  In fact, I've seen it recommended to run 2 sway bars for trailers 26 feet and up.  I added a second one, and it made a difference especially on long distance trips.  Trailers often get squirrelly in wind - far more than 5th wheels.
3) I agree with others to visit a local dealer.  Also know that comparing two trailers that seem similar may not be an apples to apples comparison.  Many manufacturers have several model lines ranging from lightweight budget-friendly entry level trailers on up to heavy luxury trailers.  Don't judge trailers just by floor plan and price. 
4) Maybe wait on a membership until after you camp a bit.  We found we like to mostly stay at state parks, but we stay at KOA parks two or three times every season, and the savings of being members pays for the membership. KOA parks are fun with lots of amenities, but they can get expensive and not everyone enjoys them.
Good luck!