Admiral Act I Objective
Prove Yourself as a New Admiral
While you're a Regent and winning an ambition, advance 
1 for each Imperial ship that anyone places on the map.
Imperial ships are placed with the Use Imperial Foundries and Govern the Imperial Reach (Policy of Escalation) edicts.
Demonstrate the Navy's Might
While you're a Regent, advance 1 for each Trophy you take in battle with attacking Imperial ships.
To advance, you can battle Blight, Free cities, and Outlaws.
You can attack with both Loyal and Imperial ships, and the Imperial ship can be destroyed.
Act: 1
Q: Do trophies taken as part of ransacking the court count towards Admiral's objective?
A: Yes, if Ransacking the Court is triggered by destroying a city in battle and not by some other means. You are destroying pieces in the battle, and Ransacking the Court happens as part of destroying a city.