Power supply and pcb connections to ground, earth
If the first metal box is grounded to protective earth then that satisfies safety and most emc requirements. Connecting the output negative line to that same ground also reduces emi from the dc output causing a problem with dc connected equipment. But, in cases where the dc output has to remain galvanically isolated from ground, the capacitor mentioned in your 2nd question will usually suffice for reduction of emi from the dc output.

That secondary side emi comes from the primary switching and is capacitively coupled to the secondary through the isolating switching transformer producing common-mode noise on the dc outputs.