No, that's not the standard, but yes they have an uphill battle.
These are facts that they won't have in their favor.
1. They didn't indicate that they fired you for misconduct.
2. They don't have documentation warning you about not doing to integral task to your job.
These are two facts that really would make a misconduct claim work. You begin with the right to unemployment and for it to be taken away, sufficient proof of misconduct has to be put forward.
Now, there isn't anything that you can do to protect yourself at this point. The facts are already in place. Anything you do at this point is happening after the fact and will not have any legal bearing on the question of misconduct at termination. It's not like there is any sort of form or letter that you can send to cut off their right to claim misconduct. You can't now send lesson plans and somehow change the fact that you didn't send them in the first place.
It is either misconduct or it is not. Based on the things that you've told me, I'm inclined to believe that they can't establish that it is misconduct, but there simply is nothing that you can do at this point about the issue. Wait to see if they fight your unemployment and if they do, THEN you'll be entitled to send some explanation of the facts to the unemployment office specifically in response to their allegations (because you'll actually have them rather than trying to guess what they might be).