With regards to the damp; stick a couple of vents in it somewhere. If there’s an overhang on the roof, a hole cut out tucked up under the overhang covered by a vent to keep the rain out will help massively. Bonus points if you can put one low down on the opposite wall to the roof vent too. The convection/chimney effect you get happening will allow allow air to circulate across the entire space getting rid of most of your condensation issue. Try keeping the vents on sides that aren’t blocked by anything so you can get a decent breeze blowing in.
Three 6″ x 2″ vents underneath the overhang and a half inch gap under the door on my garage has stopped dead any condensation I used to get in there.
If you’re putting the shed on a wooden frame you’d have to take in to account how strong the base of the shed is when not supported by slabs or concrete underneath. You don’t want to be standing in the shed and damaging the floor where it falls between the joists. I’ve never seen one of these sheds up close so I don’t know how much of an issue it would be. Putting deckboards over the top of the frame you make would solve that particular problem and not cost much time or money.
Good shout on the weed barrier.