Why does the convert have his bris before the mikvah rather than afterwards as a Jew? Seemingly the bris done before is not a mitzvah so why not have him do bris milah after the mikvah?
R David Brofsky asks and answers your question in a series of lessons on the laws of conversion and circumcision (here) and brings different opinions
The Talmud (Yevamot 47b) teaches that after the beit din informs the convert of the difficulties and hardships entailed in joining the Jewish People, the convert accepts all that has been said, and "they circumcise him immediately … When he is healed [from the circumcision], they immerse him immediately … [and] once he has immersed and emerged, he is like a born Jew in every sense."
The Rashba explains that “since the tevila embodies the essence of the conversion, leaving the impurity of being not Jewish and entering into the sanctity of Israel, therefore it [the tevila] must be at the end.”
Similarly, the Ritva writes that “when he is uncircumcised, the tevila does not work, as it is similar to one who immerses while still holding something impure (tovel ve-sheretz be-yado). Therefore, even after the fact, the tevila [performed in such an order] is invalid.”
The Ramban (ad loc. s.v. nitrape) disagrees with these Rishonim and insists that there is a different reason for delaying the tevila. The convert is first circumcised as a means of deterrence; thus, if he did immerse before being circumcised, the conversion is valid. Tosafot cites as a proof for this assertion another passage that teaches that when a pregnant woman converts, her son who is born thereafter is considered to be Jewish, and tevila is not required. Since the child is not circumcised until after he is born, clearly the order of the mila and tevila does not matter. (We will discuss this passage in a different shiur.)
The Rema (YD 268:1) cites both views. The Shakh (2) rules that one who immersed before being his circumcision should immerse again.
This debate may be rooted in a fundamental question regarding the circumcision of a convert: Is the mila a mere preparation for the tevila, such that it is the tevila that actually effects the conversion (Tosafot, Rambam, Rashba, Ritva)? Or are the mila and tevila both significant parts of the conversion process, such that it therefore may not matter which is performed first (Ramban)?
See further there for halachic implications of these opinions.