Is it permissible for a person who has to make up for missed fasts to fast six days of Shawwal (the month directly after Ramadan) before making up for the missed fasts?
Yes, it is permissible for him/her to do so, but it is better that they make up for missed fasts first. And Allah Knows Best.
Is permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.
Should Al-Basmalah (i.e. saying Bismillaaahi Ar-Rahmaani Ar-Raheemi) be recited after Takbeerat Al-Ihram (saying Allahuakbar) which marks the beginning of prayer?
All perfect praise be to Allah,The Lord of The Worlds A praying person is obliged to recite Al-Basmalah at the beginning of Chapter Al-Faatihah, and in every Rak`ah (unit of prayer) because it is a verse of it. And Allah Knows Best.
Is the patient who is on dialysis twice a week obliged to observe fasting? Is his fasting on the days he is on dialysis considered valid or not?
Praise be to Allah the Lord of the Worlds.
Dialysis invalidates fasting because of the nutritional fluid used in that procedure, as pointed by specialized doctors, in addition to causing bodies to enter the stomach. Therefore, it is recommended that the patient who had undergone dialysis to observe fasting for the rest of that day, if he is able to, and make up for it after Ramadan. However, if the doctor told him that fasting, at any time, would damage his health, then he must break his fast and pay a ransom against every missed day. We pray that Allah blesses this person with recovery. And Allah The Almighty Knows Best.