What should I do with the ground wire?
Which color (brown or blue) should I connect to the black wire in the ceiling?
The fixture still works if you reverse the wires, but the socket sleeve will be hot, and anyone who touches it while changing a bulb can get a shock. When wired correctly, the socket sleeve is neutral and only the small metal tab at the base of the socket is hot.
In regards to the ground.
if you're concerned you can drill a hole, put a screw and wrap the wire around the screw on the fixture.
Or you can wait for one of the electricians on this forum to respond.