Polukranos, World Eater

An ability that triggers when a creature becomes monstrous won't trigger if that creature isn't on the battlefield when its monstrosity ability resolves.

As Polukranos's triggered ability resolves, Polukranos deals damage first, then the target creatures do. Although no creature will die until after the ability finishes resolving, the order could matter if Polukranos has wither or infect.

If some, but not all, of the ability's targets become illegal, you can't change the division of damage. Damage that would've been dealt to illegal targets simply isn't dealt.

Monstrous isn't an ability that a creature has. It's just something true about that creature. If the creature stops being a creature or loses its abilities, it will continue to be monstrous.

Once a creature becomes monstrous, it can't become monstrous again. If the creature is already monstrous when the monstrosity ability resolves, nothing happens.

The value of X in Polukranos's last ability is equal to the value chosen for X when its activated ability was activated.

You choose the division of damage as you put the ability on the stack, not as it resolves. Each target must be assigned at least 1 damage. In multiplayer games, you may choose creatures controlled by different opponents.