Is it permissible for the mother to give her children from the Zakah (obligatory charity) of her money?
It is permissible for the mother to give her children from the Zakah of her money if they are Zakah-eligible recipients, and she isn`t obliged to provide for them.
What is the ruling on someone who perform their prayer after finishing it because they believe they missed a Rak'a, a prostration, or that they did not perform it correctly (thinking their prayer was invalid)?
If he was sure that his prayer is void then reperforming it is an obligation along with figuring out the reason of invalidity so long as this wasn't out of uncertainty. And Allah Knows Best.
What is the authenticity of the supplication: "O Allah, bring it upon us with blessings and faith, safety and Islam, and success in what You love and are pleased with. My Lord and your Lord is Allah"?
On the authority of Talhah Ibn Ubayd Allah (may Allah be pleased with him) who reported that the Prophet (PBUH) would say when he saw the crescent moon: "O Allah, let this moon appear over us with blessings and faith, safety and Islam. My Lord and your Lord is Allah."
This was narrated by Tirmidhi (Hadith no. 3451), who said: "It is a Hasan (good) Hadith."
And Allah The Almighty Knows Best.
A person bought a device through a usurious bank, and receives income from it every month after paying the bank for the price. Now the person regrets it, what should he do now?
This person has fallen into what is forbidden due to dealing with usury. He must repent sincerely, and whoever repents, Allah Will Accept their repentance. As for the device, there is no need to sell it. And Allah Knows Best