Annotated chess game

As you pointed out, attempting the Four Move Trick, or Scholar's Trick.

But if 2... Nf6 3. Qxe5+

This is ok, but not the best for the move played. Instead, 2...Nf6 was best to counter the upcoming threat. You're right. Having said that, black was right to just go ahead with his development. But, later on ...Nf6 will be the way to go for black.

If 3...Nf6 4. Qxf7#

This works but is not the best move. 3...Nf6 also develops and stops the threat again. As now, the Queen is still prone to harassment. True. This is best, but at a later time, it's useful.

The move that I recommended still works again and stops the threat.

Yes, this is where he goes wrong, as it was his last chance to develop his N to f6.

Just note that if he had played 4...Nf6, he'd have a space advantage. As mate would be stopped, the Q would be prone to harassment, and black would wind up with a space advantage because more pieces are developed. And, there's more action in black's center. So, there's only a few occasions where you'd want to bring the Q out before developing pieces. But, this is not the reason as we call this a cheapo tactic which is a threat that's easy to drive off. So, it's basically like skipping steps in math since you need more pieces to make a threat that's difficult to drive off. As multiple pieces are, generally, used in an attack. And, when the opponent finds the right moves to threats that can easily be met, they're generally better off. It's also like Math, believe it or not. That's because if a teacher gives HW or a quiz, a lot of times, if you get the answer right but skip steps, they only give you partial credit.