We have a room that had a 2 wire socket, originally installed, and a three wire socket that was changed from a two wire socket by myself, called "Socket A". Recently an electrician installed two small fans in the same room and changed the 2 wire socket "Socket B" into a 3 wire socket. He did not connect the ground on the socket to the box, saying that the socket is grounded and that is all you need.

Since that time Socket A has not worked. Could the unconnected ground in Socket B have any relationship to the failure of Socket A to operate? The sockets are on the same breaker

No ground will not stop lights from working, 3rd prong on socket will not be grounded.
It should be connected to green or bare wire, or the metal fixture?
Is this a Receptacle? Or Fixture?
Do you have green or bare wire in box?

Outlets are typically daisy-chained. Power enters the duplex receptacle and leave for the next one. A loose connection generally causes all outlets down stream to malfunction. Call the electrican back.

They are duplex recepticles and probably in serial connection. The working connection has an unconnected ground.
Should the electrician connect the ground to the case in all circumstances?

There is a bare gound wire in the incoming wiring that is not connected to the duplex outlet.

The ground wire should be connected to BOTH the metal box and the grounded outlet. Your not allowed to rely on the outlet fastening screws to make the ground connection, but this is not the cause for the outlets not working. It is evidence of poor workmanship.