Welcome and congrats on the 26BH!!!
Check all appliances, water pump, faucets, etc... Look for any signs of water damage along all walls, starting at the floor up to the ceiling. Check the roof. All drawers/ cabinet doors/ shelves/ etc... Basically everything from the coupler (tongue) to the rear bumper.
As mentioned, check the tires!!!! Trailer tires do not age that well due to all the sitting they do. It is suggested to replace trailer tires 3-5yrs.
Probably should check and go through the brakes and wheel bearings, since you don't know if the selling dealer did or the last time the were gone through.
Should be ok with the F150, but verify you are within all the specs of the tv (tow vehicle). Make sure you get a good wdh (weight distribution hitch) with integrated sway control. Reese Dual Cam, Reese SC, and the Equal-I-Zer 4 way system are all very good for the money. The Dual Cam is a little more finicky setting up than the other 2. The Reese hitch head (both the DC and SC) allow for a little more fine tuning in regards to the wd bar adjustment. And even though the dealer will set up the wdh, chances are you will need to re-adjust it once you load the trailer up for a trip. Here is a link to wdh set up
and how a wdh works