Is it permissible for a wife to take from her husband's money without his knowledge to compensate her for what she spent from her own money on the house and children, or to buy things she deems necessary for herself and her children that he considers unnecessary?

As long as you spent your money on your husband and children with the intention of getting it back, and you informed him of this, then he must return these funds. If the husband is in straitened circumstances, you should give him respite until he is able to repay. However, if he is solvent and refuses to pay, then you are permitted to take from his wealth without his knowledge, to the extent of your debt, provided that the amount does not exceed your right and you are safe from a major detriment occurring.

Regarding your husband's spending on you and the children, if he spends appropriately (according to custom), then you are not permitted to take from his wealth without his permission. However, if he is falling short in the obligatory expenses, then you are permitted to take from his wealth without his permission, to the extent of your need for maintenance according to custom, knowing that the obligatory maintenance is a sufficient amount for food, housing, and clothing according to custom.