You are correct in saying that churches of Christ are not a member of any denomination. Each congregation is autonomous and under the leadership of it's local elders, just like in the 1st Century.
A proper candidate for baptism that has been baptized for the remission of their sins is added to the church by God. It doesn't matter who performs the baptism.
What if the baptizer (as opposed to the baptizee) is administering the baptism for reasons other than the remission of sin? For that matter, what if the baptizer has never himself (herself?) been properly baptized for the remission of sin? Does it still not matter?
A proper candidate for baptism that has been baptized for the remission of their sins is added to the church by God. It doesn't matter who performs the baptism.
Yes, that is correct. churches of Christ are autonomous and there is no central governing authority. The local elders are responsible for each individual congregation. The Bible says that God adds those who obey what the Lord requests that they do to the church. Very astute. Although many members are unfaithful and are not living as they should and scripturally do not have salvation secured, those that do are in the right "body", the only body of Christ and are headed for salvation. Anyone not in the "body" of Christ and is serving in a man-made denomination can not make such a claim to salvation per what the scripture teaches.