Is the contract considered valid, and may the wife annul it outside of court, and what is the formula for annulment, if she married with the intention of tahlīl and stipulated that the husband provide housing and furnishings within one week of consummation, but he failed to fulfill the condition?

The marriage contract is valid and is not invalidated by the intention of tahlil (making lawful for the previous husband), because the contract is nullified by the husband's intention. As for the condition of providing a dwelling, it does not affect the contract because it is a right of the woman. However, if the dowry (mahr) is the bridal trousseau, the wife does not have the right to annul the contract if the solvent husband refrains from providing it; rather, she can demand it. If he is insolvent, there is a difference of opinion regarding the establishment of the right to annulment, and the more correct view is that she does not have the right to annul. However, if the trousseau is in addition to the dowry, it is a binding condition, and she has the right to annul the marriage if the husband does not fulfill it. The general principle regarding annulment is that it can only occur through the ruling of a judge, unless the spouses mutually agree to it, in which case a judge's ruling is not required.