The crusts of the bread, if in small amounts, can be eaten as is. When there's a large quantity, you can mix them and use them as a binder for ground meat. As for the reason for cutting, it's because it doesn't match the intended texture of this sandwich. Of course, it's okay not to cut them based on personal preference.
It seems you're interested, so if you'd like, you can wash it, but I think it's unnecessary if you peel off the outer leaves. If you're in Tokyo, how about going to Ameyoko to inquire?