Assuming you want a fairly big trailer for living comfort, especially in winter months/rain/etc., I suspect you'll want to move up to a heavy duty truck. If you're looking at 250s anyway, may as well check out the 350s. I think you'll be surprised at how little the difference in cost and ride quality is for a decent bump in capacity.
If you do stick with your F150, I would recommend staying in the ~24-27 ft range and bumper pull only. Unless you have a truck with the max payload package, that's what's going to limit the size of trailer you can tow, more so than the towing capacity.
I'm sure others on here can speak to what to look for in a 4 season camper or brands to look at better than I can. My Coachmen has an enclosed bottom and the furnace lines run in there. I believe that's to help keep the water lines from freezing. I have no idea how well it works as we only camp in summer.
I can't remember which brand but I saw a fifth wheel at an RV show a couple years ago with kayak storage inside of it. A fifth wheel will be more stable and nicer to tow so I would lean that way as long as you can make it work with the gear you want to bring. Another option would be to just toss the kayak inside the trailer when you're traveling then move it to the truck when you arrive at a stop.
Also, if you're looking at a small boat, most places would let you tow that on a second trailer behind the fifth wheel. There are some states that would allow another trailer behind a bumper pull but not nearly as many.
We bought our trailer gently used (~4 years old). That's the way to go, in my opinion. As long as you can find one that's been well maintained, let someone else deal with all the construction issues and take the depreciation hit. I would expect to see a lot of 2 year old trailers on the market this year as travel restrictions are being lifted and airlines add more flights.
I'm really happy with my Coachmen Freedom Express and my parents in law are mostly happy with their Forest River Rockwood Mini Lite (most of their complaints are around size and storage/layout, rather than quality).
Things may be different in the US but we have very few chain RV dealers in Canada. I would look to buy private party first but, if you decide on new or can only find what you want from a smaller dealer, I would have no hesitation to buy from an independent shop. The only reason to bring it back to a certain dealer would be warranty work. For anything else, I would find a reputable independent repair shop.