In that screen-shot the move button is supposed to behave like the 'Show advance' button in settings forms and upon clicking reveal the category dropdown + a 'Go' or 'Update' button. Then selecting a category in the dropdown and clicking the Go/Update button will move the items but also hide these two elements until next time some questions need to be moved.
But I prefer to remove this moving functionality from this section altogether. In too many cases it is not very effective anyway since one may need to go to the categories tab and create a new category to move the questions to.
In fact I think (but wanted to get to that gradually) that question bank management (including adding and deleting questions but also adding/deleting/moving categories) should be done only in the question bank and not in quiz editing. While I can appreciate that some users may want to "simply" create a quiz and its questions on the fly and on one page, I think that this is a bad practice which only results in a mess, which in turn gives these users occasion to complain that they are confused and blame the whole world but their own practices. And to think that some such users deduct points from their students' works for lack of organization ...
So, one option is to remove from quiz editing all question bank functionality except for editing of individual questions. In quiz editing the list of questions would be fixed (per category) and the control buttons allow for only adding or removing questions from the quiz. (Btw, I haven't investigated that part of the code yet, but perhaps it wouldn't be too difficult to add checkboxes to the list of questions in the quiz and a button to remove all/selected from the quiz) Then for adding/deleting/moving questions you need to open the question bank either in a different tab or on the same page but in a clearly separated section (although I can see how some users may still be confused in the latter approach).
If we want to preserve the current question bank management functionlity in quiz editing, and make sure the quiz control remain more prominent, the whole section of the question bank controls may be hidden under a button of the 'Show advance' sort. Of course, unique styling of that section may also contribute to making the question bank functionality less prominent (I haven't gone into that in my sketch but it was on my mind).
