Is it permissible for someone who has forgotten to mention the supplication pertaining to answering the call of nature to say it while in the toilet?
It is disliked for a Muslim to mention Allah while in the toilet, but he/she may recall the supplication in his/her head without uttering it; that is in case he/she had forgotten mentioning it before entering the toilet.
What are the conditions of the Aqeeqah?
The animals that are acceptable to be slaughtered for 'Aqeeqah are the same which are acceptable to be slaughtered for the sacrifice of 'Eed in terms of their age and lack of defects. The sheep should be one year old, and its teeth should have fallen off. As for goats and cows, they should have finished two years of age and entered into the third. The Aqeeqah should be free of any defects as regards meat, or price value.
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 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.