What is the obligatory Zakat on two apartments, two commercial shops, and the wife's gold jewelry, and who is responsible for paying it?

There is no zakat (definition) on the money obtained from the rented apartment as long as it is spent on housing rent, due to it not reaching the nisab (definition) and a hawl (year) not having passed. As for the other apartment: if you own it with the intention of trade, zakat is obligatory on it (2.5%) annually if its value reaches the nisab; otherwise, there is no zakat on it. The money obtained from renting shops is subject to zakat for your mother if it reaches the nisab and a hawl has passed over it. Regarding the zakat on the wife's jewelry, it is more prudent to pay zakat to avoid disagreement.