Is Zakat obligatory on a non-residential apartment, on the gold worn by the wife, and on invested capital? And is it permissible to give Zakat al-Mal to help a man repair his burned apartment?

As for the apartment you do not inhabit: if it is intended for trade, then its value is subject to zakat upon the completion of a hawl (lunar year) on the original capital. Otherwise, there is no zakat on its value, but rather on its rental income if it reaches the nisab and a hawl passes over it. As for the funds used for trade, zakat is obligatory on them if they reach the nisab (85 grams of gold or 595 grams of silver). As for jewelry prepared for adornment, there is no zakat on it according to the majority of jurists, whether it is worn continuously or infrequently, unlike that which is not prepared for adornment, on which zakat is due if it reaches the nisab. As for paying zakat to someone whose house has burned down: it is permissible if he does not have the means to repair the house, or if he has some but not all of the means. It is prohibited if he has what he needs to repair the house.