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

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

A spell or ability that counters spells can still target Pearl Lake Ancient. When one resolves, Pearl Lake Ancient won’t be countered, but any additional effects of that spell or ability will still happen.

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.