Is it obligatory to fast the days that the mother missed in her youth due to a valid Shar'i excuse, and is a kaffarah (expiation) for delay incumbent along with the fasting, given that the number of days is unknown?

The default is that your mother's religious obligations are fulfilled. However, if you are certain that she did not perform an obligatory fast, then you must feed a poor person on her behalf for every day she missed. If you fast on her behalf, it is permissible, based on Aisha's hadith: "Whoever dies and owes fasts, their guardian should fast on their behalf." It is also obligatory to feed a poor person for every day she delayed making up the fast, unless she was ignorant of the prohibition of delaying. The more cautious approach is to give two mudds: one mudd for the missed fast and one mudd for the delay, which is obligatory according to the Shafi'i school. The obligatory amount for feeding is one mudd of food (approximately 750 grams), and half a sa' (1.5 kilograms) is better to avoid scholarly disagreement. You are only obligated for what you are certain of, as the default is that there is no further religious obligation.