Is it permissible for my husband to give his brother zakat money to help him solve his financial problems, given his difficult financial circumstances, his family's needs, and his stalled home construction?

It is not permissible to give zakat to the husband's brother to complete the finishing of apartments.

If the husband's brother is poor and cannot find enough to meet his basic needs (food, clothing, etc.), and his salary does not cover these needs, then it is permissible to give him zakat due to his poverty.

It is permissible to give zakat to a student of Islamic knowledge if it is difficult for him to combine seeking knowledge and earning a living, and if he is bright, because acquiring this knowledge is a communal obligation (fard kifaya) and its benefit accrues to the Muslims.

If one of the husband's brother's sons is poor, bright, and seeking Islamic knowledge, then it is permissible to give zakat for his study expenses.