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.

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

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.