No. Ceramic cannot be cut cleanly on site; it will shatter or leave a jagged, dangerous edge.
Most pedestals have an open back. You can reach behind slightly, but generally, the design is meant to sit close to the wall so dirt doesn’t get behind it.
Yes, a bottle trap is standard. Just ensure the pipe length fits inside the hollow cavity of the pedestal.
It is usually fixed with silicone or a specialized floor screw bracket to stop it from being kicked out of position.