Is the husband obliged to pay his divorced wife the price of the furniture he sold after their divorce, especially since she damaged some items before leaving, and the furniture was owned by the husband and was not agreed upon as part of the Mahr (dowry) or marriage contract?

As long as the furniture is your property and was not part of the dowry, your wife has no right to demand anything from it, unless you donate it to her willingly. You are not obligated to pay her its value. In case of dispute, the Sharia (definition) judge will rule on the matter.