Is it permissible to give my zakat to my grandson who is studying at university, while I am living abroad and my money is in my home country? Also, is it permissible to assign my father the responsibility of distributing the zakat to the people in my home country?
It is permissible to give zakat to a son whom his father is supporting if he is an adult and in good health, because his father is not obligated to support him, making him one of the poor among the Muslims. However, the student who may receive zakat is one who is diligent in seeking beneficial knowledge for the Muslims and is religious. Zakat should be paid in the country where the money is located, and appointing your father to distribute the zakat in the country where the money is is the correct approach. And Allah Knows Best.
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 ruling on making a false oath?
Making false oath indulges its maker in Hell; therefore, he is obliged to repent, ask Allah for forgiveness and make an expiation; feeding 10 poor or providing clothes for them and in case of incapability of doing any of latters he should fast for 3 consecutive days. And Allah Knows Best.
I prayed all the obligatory prayers in congregation, but after finishing the `Isha prayer, I remembered that I had forgotten to pray Dhuhr. What should I do, and is my prayer invalid?
Your prayers are valid, but you must make up the Dhuhr prayer as soon as you remember it, based on the saying of the Prophet (peace be upon him): "Whoever forgets a prayer must pray it as soon as they remember it; there is no expiation for it except this." [Muslim]. And Allah knows best