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 having brotherly ties between a strange man and a strange woman, and is the former considered a Mahram (unmarriageable) to the latter?
Such a relation between a strange man and a strange woman is forbidden in Islam because it involves forbidding what Allah has allowed by marriage, and allowing what Allah has forbidden such as looking and the like. Such a relation doesn`t render any act lawful between them, thus they are forbidden to look at each other, have a Khulwah (seclusion), and travel together.
What is usually said upon the slaughtering of the Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth)?
It is desirable to say: "Bismillah Alrahman Alraheem, was`salatu Wa s`salamu Ala Sa`idenah Mohammad, O Allah, this Aqeeqah is from You and to You on behalf of ….."
Should vowed fast be offered in consecutive days, or not?
Vowed fast should be offered in line with what was originally intended by the vow-maker, either consecutively, or not.