So far, most everything stated is "on track".
I don't remember how many Mikvah (baptism) pools there were outside of the Temple, but it is (or was) a goodly number. As noted, there were (are) a number of reasons for undergoing a Tevilah (ritual cleansing). As some examples, after touching a dead person or unclean object, recovering from various illness, and most women undergo a Tevilah on a monthly bases and after childbirth. The practice started as a formal ritual at Mt. Sinai, prior to the giving of the Torah to the nation of Israel - and the practice continues among the Jews of today. In many scriptural references, the practice of baptisms is simply referred to as one of the several forms of "to wash".
Jesus had to under go a Tevilah in order to become ritually clean, prior to taking up His ministry as God's personal emissary (apostle) to the world. No one, except John's disciples, was baptized "into" the name of John (which is different than being baptized into John or into Christ as a disciple, or "the taking on of the name of" a rabbi). Rather, people were baptized in (by, through) the name of the authority in which the baptism is being performed, such as it was "John's baptism".
A Tevilah is performed in the name (authority) of the one overseeing (authorizing) the process. John baptized in his name (authority) as a prophet, for the remission of sins before God, not "into" his name as gaining more personal students (Talmidim, disciples). Paul was glad that he baptized only a few, so no one could accuse him of baptizing people in his own name (authority), or "into" his name, as a rabbi.
When Gentiles are baptized, we do so by the authority of Jesus Christ (
Galatians 3:26-29), and we are indeed baptized "into Christ", as both converts to Judaism (in general) and disciples of the Rabbi Jesus Christ (in particular), as noted by another poster (also see
1 Peter 3:18-21).
In the performance of a baptism, no one lays hands on another. The one "performing a baptism" is only a witness of the event! If there is a laying on of hands at all (
Acts 8:17) it is done after the baptism (
1 Tim 5:22). Why? Because when one lays hands on another prior to a Tevilah, the "uncleanness" of the other is transferred upon the one doing the laying on of hands. Or, in another mode, as when the High Priest lays his hands on the scapegoat at Yom Kippur, he is transferring the sins of Israel onto the goat that is to be led out into the wilderness. The laying on of hands is a serious matter (
James 5:14-15,
Acts 8:14-17 &
1 Timothy 4:14).
This is a very quick overview and it misses much of the history, spiritual significance, social and religious traditions, and scriptural requirements concerning the how, why, when, and where of a Mikvah and the performance of a Tevilah. It is some of the Torah (Law) that we continue to practice (go through a ritual) without knowledge or understanding. Which, I also confess I am still not well versed in, but I am studying.
I hope this contributes a little toward understanding the original question.