Just to clarify.
1- Regarding Jewish circumcision: there is no source and no reason to say that Jews made it differently in the past, and changed it to cutting the whole foreskin in Roman times. But a Jew, especially the Rav, can explain it better.
2- Regarding Christians, there are many Christian branches, so it's difficult to say how they interpret it. I am not a Christian, but my interpretation of the NT is that religious circumcision is forbidden. In fact the NT theoretically goes beyond Torah in demanding to keep the 613 mitzvot. While Hallacha states different levels of punishment for breaking a mitzvah (lashes for eating Taref, execution for kidnapping, just offering a Korban in case of swearing falsely, and so on), the New Testament states that whoever breaks one mitzvah is guilty of breaking all Torah. However the same NT states that Christians do not need to keep the Torah for two reasons:
Most Christians are not Israelites. Jesus kept the Torah for all and those who believe in him are free from many ritual commandments thorough his sacrifice. But if someone gets circumcised he loses Jesus benefits to exempt him from the mitzvot, and at the same time loses the exemption he had for being a Gentile, and so it binds him to keep all Torah. If he doesn't, he is considered more a sinner by the NT than by Hallacha, since the NT holds that breaking one mitzvah equals breaking all.
Apparently the NT thinks that just a circumcision for religious purposes makes you automatically Jewish. (the same opinion is hold by some Karaites- that if a Gentile got circumcised with intention of keeping the Torah, he must keep it).
However I don't think Christians would forbid circumcision made for medical reasons.
3- Regarding what Torah says about circumcision for Gentiles, it is clearly stated that only Abraham and his descendants need it. There are some non- Jews who descend from Abraham. The Rav could explain it.
But other Gentiles are not requiered to do it. If they want, are they allowed? I think that Bris milah (Ritual Jewish circumcision) is definitely not allowed to a Goy, since its a Jewish mitzvah.
Just a simple circumcison? Many Goyim do it for cultural or medical reasons. Is it ok? I think a Goy is only allowed if he needs it for health reasons. Not just cultural practises. Why didn't Abraham, who kept all the Torah before it was given, get circumcised at a younger age? Some say that, since Abaraham was still bound to the Noahide Laws, he could not cut his flesh unnecessarily. So he waited until Hashem ordered him to do so.