Will a house become "invalid" if I cage it to to stop an NPC leaving?
No, this will not make the housing invalid. You could completely enclose the Nurse, as long as at least one tile is exposed to join the enclosure to the main room. But one problem you might have, is that placing blocks may change the location the Nurse stands at night, so if you move away from the arena (in order to fight the pillars per say) she may no longer be standing in the same location. - George Willcox in comments