You can get away, and the engine recommends this, with chasing the queen for about 2 or 3 moves depending on the position. After that, focus on developing your pieces, and then return for a second strike on the queen. Opponents who bring their queen out early often forget to check escape squares after the first chase.
What should I do after an opponents failed scholars mate?
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You should crush them.
You actually want to focus on attacking the enemy queen by developing your pieces while doing that. This way they have to move their queen, and you are developing your pieces. Most likely you will not win the queen but that is rarely the goal in punishing early queen moves. Also your opponent's lack of pieces that are developed is a weakness in their position that can be exploited.
On another note the first move when someone is trying to play scholars mate they play e4 or e3 first. d3 and d4 don't allow for scholars mate right away, and d3 actually gets in the way of scholars mate.
Here is a Youtube video on how to deal with early queen attacks: https://youtu.be/8vJic_mR76o?si=Fs8DFbTWwfPyK1FH
Okay so sure, the opening is fumbled, but the second option they have is the similar mate, but with the knight…whatever.
Obviously focusing on killing their queen is a waste since it can move anywhere quickly.
Only thing I can think of is making sure my pieces are protected and to look for weak spots in their pieces…problem is they sometimes don’t even attempt to move many, if any pieces out unless it’s for a quick mate…?