You don't have to choose a target for Immersturm Predator's triggered ability. However, if you do choose a target and that card is an illegal target as the ability tries to resolve, the ability won't resolve and none of its effects will happen. In that case, you won't put a +1/+1 counter on Immersturm Predator.

You can activate Immersturm Predator's last ability even if it's already tapped.

Immersturm Predator's triggered ability doesn't allow you to tap Immersturm Predator. You have to find some other way to tap it. Let us suggest attacking.

For the triggered ability to trigger, Immersturm Predator has to actually change from untapped to tapped. If an effect attempts to tap it while it is already tapped, the ability won't trigger.