I think you need to know the weight of the mower. As well as how high your roof and the door should be to ride it into the shed. That should determine the floor construction. The floor joists, if not sitting on solid base, will be on the perimeter beams, and neither the joists nor the beams should bend under the weight-over long term. I am imagining pacted gravel base, extending 1-2 feet enclosed by pt rail fence, 4x4 pt floor frame (maybe 12" apart), and 2 layers 3/4" pt ply. I don't know what your mower needs for a floor. Its wheels will be on the floor, most likely over 4 points on the floor plywood, so the weight will be concentrated over small areas, not evenly over the entire floor. You also may want concrete slab. Plywood could become quickly damaged if the mower drips fluids or water.
You may want to post this question in lawn section for real life experiences.
For such a shed, wall spacing of 24" and 3/8" ply is fine. But closer spacing and 1/2" ply min is better because whatever cladding you use, there will be less waste as well as more nail holding capacity. If you can plan to nail into the studs, you may skip the underlayment.
If you want 2:12 roof, it's better to use the roll roofing. For small shed, you can nail the perimeter instead of gluing the whole roofing. All nails will go through the roof deck. You may hit your head on the nails if the roof is too low. The rolls also come in self stick.