The bench needs to be rigid. Consider panelling the back and ends, to increase torsional stiffness.
A shelf can be used to store tools, chucks, faceplate etc, and will add to rigidity.
The lathe needs to be “leveled” to take any twist out of the bed, so the feet will need to be adjusted / shimmed. Adjusting the bench feet won’t be much good, but it should not wobble on the floor.
My preference is for riser blocks, since a screw adjuster is more precise than shimming.
As already said, ensure that the shop is insulated, but has a small amount of ventilation to minimise the risk of rust (And who wants to work in a cold workshop!)
The ventilation only needs to be a small hole (suitably guarded against ingress by spiders and bugs), say 1″ diameter, at floor level, to let out moist air.
Security
A good solid bolt on the door, using coach bolts with the domed heads on the outside, and a good hardened padlock.
You can make hinge bolts by turning the heads off a couple of woodscrews. Use a drill chuck to screw them into the back of the door. Trying to close the door will mark the frame where the holes need to be drilled for the hinge bolts to enter.
In this way, the bolt and padlock will secure one end of the door, and the hinge bolts will prevent the door being lifted out even if the hinge pins have been driven out.
HTH
Howard