IEC 61000-4-5 2 kV chassis ground to neutral and live surge DC
connect PE with the metal case is OK, if not a must (would need to lookup the standards...)
Connect the DC GND to PE - can be done. Like here https://en.wikipedia.org/wiki/Extra-low_voltage#/media/File:Extra_low_voltages_(third_safety_class).JPG But you should add  capacitors in parallel with varistors (or other suited surge protectors) from DC+ and from DC-GND to PE. If you connect directly huge currents can be an problem
Shields are also only connected with PE (via resistor in parallel with a capacitor at both ends, or with a capacitor only at one only) - you do not want DC current flowing in the shield, the resistor is only to provide a path, so it is not floating.
And AC neutral line is no GND and normally uses can plug in the plugs to sockets reversed, so suddenly Net's L1 suddenly is your neutral and you won't that connected with your DC-GND