Can I filter the backlog items shown in the "Backlog items that are not assigned to any sprint" box?

Yes, if you create saved filters in the Backlog tab, these filters will also be available for use in the "Backlog items that are not assigned to any sprint" box in the Sprints tab:

Tip: Make sure you save the filter, by clicking "Save this filter" inside the filter box in the Backlog tab. Otherwise, it won't appear in the "Show" dropdown.