Does the current failure to pay zakat—due to needing time to understand it and to avoid a mother's anger given financial circumstances—affect the validity of fasting, knowing that there is an intention to pay it later?

Zakat is one of the pillars of Islam. It is not permissible to delay it beyond its due time for one who is able to pay it and finds someone deserving. You must pay Zakat on your own money even if your mother objects, because there is no obedience to a creature in disobedience to the Creator. Zakat must be paid for all past years in which you did not pay it, calculating the amount due for each year. Debts are deducted from the money before paying Zakat if the remainder reaches the nisab and a full year has passed. Not paying Zakat does not affect the validity of fasting. Zakat is not obligatory on money you do not own, and its payment is not valid except with the owner's permission.