Cycling is an activated ability. Effects that interact with activated abilities (such as Stifle or Rings of Brighthearth) will interact with cycling. Effects that interact with spells (such as Remove Soul or Faerie Tauntings) will not.

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.

The cycling ability and the triggered ability are separate. If the triggered ability is countered (with Stifle, for example, or if all its targets have become illegal), the cycling ability will still resolve and you'll draw a card.