Any Equip card that doesn't specifically say otherwise, can be equipped to your opponent's monsters. Even if that would be disadvantageous to you, it still can be done. It also means if a monster is equipped with Axe of Despair, it still remains equipped with it even if control of the monster changed.
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I run a very good dragon deck and the only equip card I run is Axe of Despair. 1000 attack.
It will still be treated as '0' except for the purposes of further addition or subtraction. That is to say if a monster's ATK was reduced to -500, then it will be treated as '0' for the purposes of Damage Calculation. But if Axe of Despair is equipped to it, then its new ATK will be -500 + 1000 = 500ATK in total.
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In order to make lumber in Harvest Moon: Tale of Two Towns, you need to have the axe. When you find branches around the mountain area or in all the hidden areas, just equip your axe and chop the wood into lumber.
A small axe is simply called "a small axe" or a "hatchet".