Once a ward ability has triggered, it doesn't matter if the artifact loses ward (most likely because Bronze Guardian leaves the battlefield in response) before the ability resolves. The controller of the spell or ability will still have to pay {2} or the spell or ability will be countered.

If a player casts a spell that targets multiple permanents their opponent controls with ward, each of those ward abilities will trigger. If that player doesn't pay for all of them, the spell will be countered.

The ability that defines Bronze Guardian's power applies in all zones, not only while it is on the battlefield.