I don't know what he did to fry the multimeter. Most probably he had selector in wrong position. Sad.
If I understand it correctly, he disconnected the earthing from from each apartment coming to the earthing strip at the ground level. Then he connected one probe of the bulb to the main neutral wire and checked earthing wire coming out of each apartment using the other probe. In that case there was no wire going to the ground right? It's just an wire coming out of each apartment,which is not connected to earthing.
In that case bulb should not glow at all as there is a open circuit unless there is leakage from live wire to the body of the appliances. It looks like three is leakage in 4 houses. One way to check is turn of the mains to those 4 houses and check again.
Did you check the color of copper wires while disconnecting. If there was huge leakage, the wire gets charred and brittle.
If you have option to get a separate earthing done for your apartment, that's a better option in my opinion. You will be saved from such stray leakage from other people's appliances.