Can an idol worshipper perform a brit mila on a Jew?

If no Jew is available, is it preferable for a Samaritan (Cuti) or an idol worshipper to perform it? What are the sources for the various opinions, and which authorities support each view?

Is brit mila required to be performed lishma—specifically for the sake of the mitzva?

Two different verses are cited to explain why an idol worshipper may not perform a brit mila on a Jew. What is the practical difference between the two interpretations? The Gemara offers three possible distinctions, though the first two are ultimately rejected. The remaining practical implication concerns whether a woman is permitted to perform a brit mila.

Can an idol worshipper perform a medical procedure or prescribe medicine for a Jew? Under what circumstances is it permitted? Is there a difference between an idol worshipper and a heretic, and if so, why?

Why did Rabbi Yishmael not permit Ben Dama to be healed by a heretic? If the Torah says "and you shall live by them," why wasn't healing allowed in this case? Rabbi Yishmael prohibits transgressing idol worship and other commandments publicly, even under threat to life. This approach differs from Rabbi Akiva's opinion in Sanhedrin, which holds that one must give up one's life rather than transgress the three cardinal sins: idolatry, murder, and forbidden sexual relations.