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If, during its controller's declare attackers step, Mage-Ring Bully is tapped or is affected by a spell or ability that says it can't attack, then Mage-Ring Bully doesn't attack. If there's a cost associated with having it attack, its controller isn't forced to pay that cost. If they don't, Mage-Ring Bully doesn't have to attack.

Once it triggers, prowess isn't connected to the spell that caused it to trigger. If that spell is countered, prowess will still resolve.

Prowess goes on the stack on top of the spell that caused it to trigger. It will resolve before that spell.

Any spell you cast that doesn't have the type creature will cause prowess to trigger. If a spell has multiple types, and one of those types is creature (such as an artifact creature), casting it won't cause prowess to trigger. Playing a land also won't cause prowess to trigger.