Praise be to Allah.
Firstly: delaying making up missed fasts until one is unable to fast
If someone delays making up missed fasts until he becomes unable to fast, he must feed poor persons in compensation, but if there was no excuse for the delay, he must repent. As to whether he also has to offer expiation for that delay, there is a difference of scholarly opinion concerning that.
It says in al-Majmoo‘ (6/366): Section: Difference in scholarly views regarding one who delays making up missed Ramadan fasts with no excuse until the following Ramadan begins:
We have stated that our view is that he must fast the present Ramadan, then make up the missed fasts from the earlier Ramadan, and for each day he must offer a fidyah (penalty), which is one mudd of food. This is the view of Ibn ‘Abbaas, Abu Hurayrah, ‘Ata’ ibn Abi Rabaah, al-Qaasim ibn Muhammad, az-Zuhri, al-Awzaa‘i, Maalik, ath-Thawri, Ahmad and Ishaaq, except that ath-Thawri said: The fidyah is two mudds for every day.
Al-Hasan al-Basri, Ibraaheem an-Nakha‘i, Abu Haneefah, al-Muzani and Dawood said: He must make up [the missed fasts], but he does not have to pay any fidyah. End quote.
This fidyah is not multiplied by the number of years that have passed since delaying making up the missed fasts; rather it is paid only once, for each day of which making up the fast was delayed with no excuse until the following Ramadan began.
Secondly: son giving his zakaah to his mother to feed poor people if she cannot afford to do that
If the mother is unable to feed poor people, it is permissible for her son or anyone else to give zakaah to her, because that comes under the heading of paying off a debt, and it is permissible for a son to give zakaah to his parents to pay off their debts, as it is not obligatory for him to pay off their debts, even if it is obligatory for him to spend on their maintenance.
An-Nawawi said in al-Majmoo‘ (6/229): Our companions said: It is not permissible for a man to give to his child or father on whose maintenance he is obliged to spend anything of that which is the share of the poor and needy, for two reasons: the first reason is that he [his child or father] is independent of means as a result of his spending on him, and the second reason is that by giving that to him, he is benefitting himself, by cancelling out the obligation to spend on him.
Our companions said: It is permissible to give his child or father some of the share of those who work to collect and distribute the zakaah, slaves with a contract of manumission, debtors and those who are fighting in Allah’s cause, if they fall into any of these categories. End quote.
It says in al-Mawsoo‘ah al-Fiqhiyyah (23/177): The Maalikis and Shaafa‘is, and Ibn Taymiyah among the Hanbalis, restricted the giving that is prohibited to the share of the poor and needy. But if a person gives his father or his child something of the share of those who work to collect and distribute the zakaah, slaves with a contract of manumission, debtors and those who are fighting in Allah’s cause, there is nothing wrong with that.
They also said: If he is not obliged to spend on him [his father or child], it is permissible to give zakaah to him. End quote.
The word translated here as debtor refers to a debtor who cannot afford to pay off his debt, whether his debt is owed to another person or to Allah, may He be exalted.
It says in Kashshaaf al-Qinaa‘ (2/282): It is permissible to receive zakaah in order to pay off a debt owed to Allah, may He be exalted, such as expiation (kafaarah) and the like, just as it is permissible in the case of a debt owed to another person. End quote.
And Allah knows best.