The second triggered ability targets an untapped creature. If there are no other untapped creatures when Ana Battlemage enters, it must target Ana Battlemage. If the target becomes tapped by the time the ability tries to resolve, the ability won't resolve.

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

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."

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

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 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.