Commentary for Avodah Zarah 51:3

This story involves the thieves of Babylonia. In the story, the thieves pay a price when Rav Menashe puts them under the dreaded ban.
In the end, the Bavli notes how much greater the generally violent robbers in the land of Israel are than the mere thieves in Babylonia. The robbers praised R. Akiva for his sound advice, whereas the thieves cursed R. Judah.

A Jewish woman should not act as midwife for a non-Jew for this would abet idol worship. In the Talmud it is explained that this is only prohibited if the Jewish woman works for free. If she is paid for her work it is permitted. A non-Jewish woman may act as a midwife for a Jewish woman.

A Jewish woman may not act as a wet-nurse for a non-Jewish child. This is for the same reason that she may not act as a midwife.
A non-Jewish woman may be a wet-nurse to a Jewish child, provided she nurse the child on the premises of the Jewish family. The Jewish family may not give over their child to the non-Jewish woman for fear that she will kill the child.

This baraita expands on the mishnah, explaining it, limiting its applicability and bringing in a dissenting opinion. According to R. Meir, the idolatrous woman may not serve a midwife for a Jew even if others are watching her.

R. Meir rules that even if the idolatrous woman is being watched, it is still prohibited for her to act as a midwife because a midwife can always find the opportunity to murder the newborn, especially by pushing in his softspot (I was always freaked out by the soft spot with my own kids).
The Talmud illustrates this with a story whereby a woman, seemingly a midwife, brags how many Jewish children she has killed.
The rabbis read her words as an exaggeration. She’s just showing off by exclaiming how many Jewish children she has killed.

This baraita is the same as that which we saw yesterday concerning midwifery.

R. Meir does not allow the non-Jew to nurse the Jewish child even if others are watching because she still may find the opportunity to kill the child.

The Talmud now explains why we need both baraitot. If we only had the source about the prohibition of midwifery, we might have thought that the rabbis agree with R. Meir that it is prohibited for a non-Jewish woman to nurse a Jewish child, because she might be able to surreptitiously kill the child. The sages would permit midwifery because it is easier to supervise.

The Talmud now explains why we need both baraitot. If we only had the source about the prohibition of midwifery, we might have thought that the rabbis agree with R. Meir that it is prohibited for a non-Jewish woman to nurse a Jewish child, because she might be able to surreptitiously kill the child. The sages would permit midwifery because it is easier to supervise.

The baraita cited here allows a Jew to act as a midwife for a non-Jew, as long as she is paid for her services. This seems to contradict the mishnah categorically prohibits a Jewish woman for acting as a midwife.
R. Yoseph solves the difficulty by claiming that this was permitted to prevent enmity. In other words, in principle it should be prohibited, but it is permitted because such a law would create enmity between Jews and non-Jews.

R. Joseph here thinks that a Jew should be allowed to act as a midwife on Shabbat for a nonJew even though this would involve desecration of Shabbat. This would be to avoid enmity. Abaye remarks to him that the Jew could offer an excuse so as to avoid enmity without having to actually desecrate Shabbat. As we shall see, Abaye offers the same type of comment in each of the following cases below.

Abaye here provides excuses for how a Jewish woman can avoid enmity and still not nurse the child of an idolater.

The source quoted here is a pretty harsh and violent source. The baraita states that while a Jew may not actively kill non-Jews and shepherds (who are considered low life thieves), one is not obligated to save their lives either. R. Joseph thinks that as long as one is paid to do so, it should be permitted. But Abaye says that one should offer a flimsy excuse rather than save their lives. I must admit that I find it hard to believe that such an excuse would avoid enmity. “Sorry, I can’t save your life because I have to go help my son off the roof.”

There are types of people who may actively be murdered. It is hard to know exactly who these people are, especially what the difference between “heretics” and “apostates” is. There are also other versions of this list. “Informers” seems to be akin to what we call “traitors” a crime that is still considered extremely serious and for which one can be executed or jailed for an extremely long time.