To easily defeat the alligator, use the Clap Of Thunder.

Repeatedly throw bombs at him until he splits into two. Then, attack them until he turns into the big invincible form. Repeat to win.

-To get the Mortician statue, go to the Prison Cliffs and cross the first broken bridge. Go down where it had broken and get the golden trophy.
-To get Lord Beckett's statue, go to the last part in Port Royal where you find a switch in one area to get to the next. The trophy is behind it.
-To get the one on the Cannibal's Island, go to where the water spot is found.
-To get the one at the swamp, enter the town area after completing the river section. When you are heading to the crocodile, it should just be in a small square area.
-The one in Tortuga is after you go down the wooden staircase.
-The one in El Cruces is where you see Norrington running, at the far right of the area.

Activate the following cheats by pressing the corresponding sequences of buttons during gameplay in Single-Player mode:

CHEAT FUNCTION - CHEAT CODE
Invincibility - UP, DOWN, LEFT, RIGHT(5), LEFT, RIGHT, UP, DOWN, LEFT, RIGHT, UP(5), LEFT.
Infinite power - UP(2), DOWN(2), LEFT, RIGHT, LEFT, RIGHT, L, R, SELECT.
Restore health - Y(2), SELECT, LEFT, RIGHT, LEFT, RIGHT, LEFT.
Restore savvy - X(2), SELECT, UP, DOWN, UP, DOWN, UP.
Spawn axe - LEFT, L(2), DOWN(2), LEFT, UP(2), DOWN(2).
Spawn executioner axe - RIGHT, L(2), UP, DOWN, UP, RIGHT(2), LEFT(2).
Spawn pistol - DOWN, L(2), DOWN(2), RIGHT.
Spawn rifle - LEFT, L(2), UP(3).
Spawn blunderbuss - DOWN, L(2), DOWN(3).
Spawn chicken - RIGHT, L(2), UP, DOWN(2).
Spawn pig - RIGHT, R(2), DOWN, UP(2).
Silly weapons - Y(2), X, Y(3).
Seasickness mode - X(2), Y, X(2), Y.
Ghost Form mode - Y, X, Y, X, Y, X.
Gain 10 gold - RIGHT, X(2), RIGHT, LEFT.
Fast music - Y, SELECT, Y(4).
Slow music - Y, SELECT, X(4).
Disable cheats - X(6).