Is it permissible for the husband to break his promise to his friend regarding renting out the house without consulting his wife? And is the wife obligated to contribute to household expenses and help her husband with her private money, especially if he took her gold with the intention of returning it but was unable to do so?

If you own the house in partnership with your husband, he is not permitted to rent it out without your permission. Fulfilling a promise is desirable. Your participation in household expenses is an act of kindness on your part, as the husband is responsible for the maintenance, and he is not allowed to compel you to pay for expenses. What your husband took from your gold is considered a debt upon him unless you forgive him.