Is it permissible to give out Sadaqah from household expenses without the husband's knowledge, given that the husband pays Zakat but does not meticulously verify its amount, so it might be more or less than the obligatory sum?

The wife must be careful not to fall into the forbidden when spending from her husband's money. She is not permitted to do so without his permission. She can refer to Fatwa no. 34552. However, if the money is her own personal property, she may give charity from it without her husband's permission. She can refer to Fatwa no. 57536.

Regarding Zakat, if the money belongs to the husband, the wife should inform him of the obligation to pay the full Zakat, and that it is not permissible to withhold any part of it. She can show him Fatwa no. 59985, which warns against not paying Zakat. However, if the money belongs to the wife, she must oversee its Zakat or appoint someone to pay it in full on her behalf.