5/8 or 3/4 osb is good enough.

3/4" in most cases, depends on how much weight you plan to put in there. Cross bracing with 2x4 laid flat will increase the load bearing and keep the floor from warping.

Answer depends on how much weight you intend to put in the shed. Many sheds have an accessory for purchase - a metal floor which looks like a grid, which you can install the plywood or OSB using drywall screws. If the spacing between is 16" or 24" on center, then 1/2" ply or OSB should be okay for loads of about 40 or 50 lbs / sq ft. For greater loads, use 5/8" or even 3/4" plywood.

Another tip is to paint any wood that is close to the ground with an termite & fungus control material, such as Termin-8 (copper-napthanate solution.) I've had untreated wood floors fail after only a few years.

Good luck.

The thickness is determined by the load it will have to bear.The greater the load the thicker the floor should be.In most cases the average home owner can use 3/4 inch since they will not have a huge load bearing need.In my case I would need more since I store bricks and other heave objects in my shed. You can cross brace all you want alas when you go to place the load on your floor it can poke a hole right through the plywood if not thick enough.Cross bracing is for spreading the weight from one floor joists to the others it is not for spreading the weight on the plywood. Only thicker plywood will increase the strength of the plywood.I have seen holes punched right through plywood when the weight is concentrated in one place. More than one roofer has learned the lesson of walking on plywood sheathing that is just to thin to support them.