Is it permissible for a wife to take from her spendthrift husband's money without his knowledge to spend on the essential needs of the household, and is she obligated to inform him of that?

The default rule is that a wife should not take from her husband's money without his knowledge and permission, with the exception of what suffices her and her children according to what is customary if the husband is stingy with it, based on the hadith of Hind bint Utbah.

If the husband is extravagant, he should be advised, and nothing should be taken from his money except with his knowledge. However, if his extravagance leads to a deficiency in household necessities, such as clothing, then the wife may take, without his knowledge, what suffices her and her children according to what is customary.

Therefore, your mother is not permitted to take from your father's money except with his permission, apart from what has been mentioned.