Lullmage Mentor

• WWizards of the Coast2009-10-01
  Official Ruling
  A spell or ability counters a spell only if it specifically contains the word “counter” in its text. If a spell or ability you control causes all the targets of a spell to become illegal, that spell doesn’t resolve but is not countered.
• WWizards of the Coast2009-10-01
  Official Ruling
  Lullmage Mentor’s second ability will cause its first ability to trigger.
• WWizards of the Coast2009-10-01
  Official Ruling
  Since the second ability doesn’t have a tap symbol in its cost, you can tap creatures (including Lullmage Mentor itself) that haven’t been under your control since your most recent turn began to pay the cost.

Rulings via Scryfall