Technically I think the answer is yes, Fire Foxes is removed from battle, but once a character dies by the numbers you immediately move to battle resolution and can't play any other cards or activate other abilities until the battle is resolved. For instance, I can't let my Evil Character die by the numbers and then play Christian Martyr to discard the Hero. I would have to play CM while my EC is still in battle.
With Spears, however, you can use the first part to target an EC in battle before Fire Foxes dies because you don't move to battle resolution until both players agree to do so. So if I attack with David, the Shepherd holding Spears, search for a Hero, and my opponent blocks with Fire Foxes, I can discard Spears to negate and decrease FF before FF dies by the numbers.
