Is zakat obligatory on a sum of money in the bank if a year has passed, in addition to the wife's gold, if their total reaches the nisab, or on one of them? And is zakat due on the monthly salary?

Among the conditions for zakat to be obligatory are the completion of the nisab (minimum threshold) and full ownership. If the amount you possess in the bank does not reach the nisab except by adding your wife's gold, then zakat is not obligatory upon you. As for your wife's gold, its zakat is due from her, not from you, and if it is prepared for personal use, then there is no zakat on it according to the majority of scholars. The amount of the nisab is 85 grams of gold or 595 grams of silver, and what is obligatory to be given out is one-quarter of one-tenth (2.5%). Zakat is calculated by the Hijri year. For zakat on saved salary, there are two methods: either you make each salary have an independent hawl (lunar year), or you pay zakat on all the money you own when one hawl has passed since you first acquired the nisab from it, which is the easier method and more beneficial for the poor.