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 is countered by the game rules, not the spell or ability you control. For example, if you control Lullmage Mentor and cast Doom Blade to destroy a creature targeted by Giant Growth, you do not get a Merfolk token when the game rules counter Giant Growth.

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.