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.
Sigil of the Nayan Gods's ability counts the number of creatures you control, regardless of who controls the creature the Aura is enchanting. If you control the creature it's enchanting, the bonus will include that creature too.