16mm dia screw will be fine.

The rails I'd use 20mm.? Not that 15mm won't handle the loads they easily will BUT the bearing sizes are tiny and the small size creates issues with clearances for the ball-screw. With 15mm you end up having to put stand-off blocks between the plate and rails/bearings to create the clearance for the ball-screw and Ball-nut.
Even with larger and taller 20mm you still have to create some clearance for the nut, usually by machining a shallow channel in F & R plates but it's much nicer and neater way IMO. The longer 20mm bearings also provide a bit more support has well.