Is it permissible to give lessons to mixed students of the seventh grade?
It is impermissible to give private lessons to mixed students since such an act leads to serious consequences for the students in this grade are teenagers.
Is permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.
What is the stance of Ahlus-Sunnah wal-Jama'ah regarding the historical figure known as 'the Sufyani,' who some say will appear at the end of times? And what is the stance of Ahlus-Sunnah wal-Jama'ah regarding the group that killed Al-Husayn ibn Ali (may Allah be pleased with them), particularly those who directly committed the killing, severed his head, and killed members of his family?
Most of the narrations about the signs of the Hour concerning the Sufyani, who is said to appear before the Mahdi, indicate that he is the leader of the army that will invade the Kaaba, and Allah will cause the earth to swallow them. We disassociate ourselves before Allah from those who killed Al-Husayn Ibn Ali (may Allah be pleased with them), supported his killing, or encouraged it, whether openly or secretly. The inner matters are left to Allah alone. And Allah The Almighty 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.