Is it permissible to distribute the raw meat of the vowed animal sacrifice amongst the poor/to offer it to them in cooked form?
It is impermissible for the vow-maker to eat from the vowed animal sacrifice, rather, he/she should distribute it as he/she had intended upon making the vow, but if the vow was a general one, without any specification then, it is better to give it as raw meat.
Is permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.
Is the Aqeeqa due on the newborn in case of dying almost a month after the delivery?
Offering Aqeeqa after a week of a newborn death is recommended.
I prayed while I was in the hospital after a bladder surgery, and my urine was connected to me and could not be disconnected that day. I prayed after performing tayammum with my finger because I was unable to stand. What is the ruling?
After Allah grants you complete healing, make up the prayers you performed with your finger to the best of your ability, at times that suit you. And Allah Knows Best.