Does the Almighty's saying: "And give women their dowries as a free gift" refer to the value of the bedroom set that the bride's family bought as a gift for her, or does it refer to the deferred dowry agreed upon in the contract? And is it permissible to pay for the room set that the bride's family bought in installments, or must it be paid all at once?

The verse discusses a woman's right to the mahr (dowry)—both its prompt and deferred portions. The man must pay it to her willingly, and there is no sin if she waives a part of it. Furnishing the bedroom is part of the obligatory mahr. If her family agrees to purchase the furnishings and write a promissory note for its cost as a debt, then that is their right. There is no reason to fear as long as you intend to fulfill it. There is no objection to settling the debt in a single payment or in installments, as long as you both agree upon it.