If it was kicked, Elvish Hydromancer's ability can target any creature you control, including itself. If it targets itself, the token that is created won't have been kicked, so the ability will not trigger again.

If a card or token enters as a copy of a permanent, the new permanent isn't kicked, even if the original was.

If you put a permanent with a kicker ability onto the battlefield without casting it, you can't kick it.

The kicker ability doesn't let you pay a kicker cost more than once.

If you copy a kicked spell on the stack, the copy is also kicked. If the copied spell is a permanent spell, the token the copy of that spell becomes when it enters is also kicked.

To determine a spell's total cost, start with the mana cost (or an alternative cost if another card's effect allows you to pay one instead), add any cost increases (such as kicker), then apply any cost reductions. The spell's mana value remains unchanged, no matter what the total cost to cast it was.

If a spell's kicker cost was paid, the spell is "kicked."