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.

Lullmage Mentor’s second ability will cause its first ability to trigger.

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.