Is it permissible to go on Hajj by means of monthly installments with a guarantee, and is Zakat obligatory on money when paying a high annual rent?

Your question contains two issues:

First: The ruling on performing Hajj with a loan: It is permissible to take a loan for Hajj if it is a good loan and not an interest-bearing one. An interest-bearing loan is forbidden, and the Hajj of one who takes an interest-bearing loan is not "mabroor" (accepted and rewarded). Imam Ahmad holds that Hajj performed with ill-gotten money is invalid and does not fulfill the obligation of Hajj. Financial capability is a condition for the obligation of Hajj. One who is unable to perform Hajj bears no sin if they die before performing it.

Second: The ruling on zakat for money saved for rent: Zakat is not obligatory on this money unless the money of each spouse reaches the nisab (minimum threshold) and a hawl (one lunar year) passes over it. The money of one spouse is not combined with that of the other to complete the nisab. If one spouse receives a sum from the other's money each month for rent, and the accumulated money in the possession of the recipient reaches the nisab, and a hawl passes over it, and there is no debt that exhausts or diminishes it below the nisab (such as the due rent), then zakat becomes obligatory on the owner of the money. If the rent is paid in arrears, there is no zakat on the savings because it is a debt owed to the landlord.