Yes, you need to equip a weapon to use its abilities. The following is from page 17 of the rulebook:
Weapons are a special type of card. A weapon adds nothing to the party and you cannot use its Dungeon abilities unless it is equipped to a hero. To equip a weapon, a hero must have Strength equal to or greater than the weapon’s Weight .
Village effects only work when you are in the village and vice-versa.
Hope that helps!