The ability of the emblem created by Kaito's last ability resolves before the spell that caused it to trigger. It resolves even if that spell is countered or otherwise leaves the stack without resolving.

You don't have to choose a target for Kaito's first loyalty ability. However, if you do and that target is illegal when the ability tries to resolve, it won't resolve and none of its effects will happen. You won't draw or discard. The loyalty counter that was added to Kaito to pay the cost of the ability will remain on Kaito, however.