When you cycle this card, first the cycling ability goes on the stack, then the triggered ability goes on the stack on top of it. The triggered ability will resolve before you draw a card from the cycling ability.

You can cycle this card even if there are no legal targets for the triggered ability.

The cycling ability and the triggered ability are separate. If the triggered ability doesn't resolve (because, for example, it has been countered, or all of its targets have become illegal), the cycling ability will still resolve, and you'll draw a card.