You are allowed to fast before them and after them- and thus you can also eulogize. And the Bach writes that on Chanukah you are not allowed to fast beforehand, and this is also the opinion of the Pri Chadash. And there are people whose custom is to fast the day before Chanukah to make up for [not fasting] Erev Rosh Chodesh [Tevet] (which occurs during Chanukah and hence forbidden). They should notc change their custom, but ideally one should not adopt that practice. And at least it is not allowed to declare a public fast for the day before Chanukah. And similarly it is forbidden to declare a public fast for the first day after Chanukah, even if an individual can fast on that day

We fast on the 13 of the month of Adar - Because in the days of Mordechai and Esther they gathered on the 13 of Adar to fight and stand for their lives and they needed to ask for mercy and they pleaded to be helped by Hashem to take revenge from their enemies. We find that on the day of war they were fasting as the Rabbis says that Moshe Rabbeinu, of blessed memory, fasted on the day [the Israelites] fought with Amalek and therefore certainly also in the days of Mordechai they were fasting on the same day; and therefore the Jews began a custom of fasting on the thirteenth of Adar. And they called this fast Taanit Esther to remember that God looks and listens to every person in their time of distress when they fast and repent to God with all of their heart just as they did in those days:

To fast on the fifth day - but on erev Shabbos you don't schedule/set a fast day because of the honor of Shabbos since usually we say Selichot and pray for mercy and on Erev Shabbos its not proper to do so because you don't have time to prepare for Shabbos. And someone who is traveling and he forgot that [one is supposed to] fast on the fifth day and he ate and at night when he arrived home and he was notified that it was a fast day he should fast on Erev Shabbos. The Maharil did not teach halachah on Ta'anis Ester because everyone was busy preparing for purim, unless if purim came out on Sunday then he (the Maharil) would teach on Ta'anis Ester (which would then fall out on the previous Thursday, because there would be enough time on Friday to prepare for Purim). The Elya Rabbah wrote "it appears to me that (the Maharil's behavior) only applied to attending. Yeshiva but regular study in the Beis Midrash was not included because this does not infringe on people's ability to prepare properly for the Seudah". We don't say the prayer Tzidkascha on Shabbos when Purim falls on Sunday, just like we don't say Tachnunim on Erev Purim that falls on a weekday likewise we don't say Tziduk HaDin on Shabbos that is Erev Purim.

In pain - This refers to someone who has pain in his eyes. But someone who is pregnant shouldn’t fast even if they are not in pain, so says the Yeshuas Yaakov. And the Elya Rabba is more strict on the matter (i.e. he says if you're pregnant you should fast), except if it's within a month of the woman giving birth, he also is lenient.

And repay afterwards - According to the aforementioned Yeshuot Yaakov, it would seem that this only applies to individuals suffering eye pains. [i.e., the requirement to repay for the missed fast of Taanit Ester] that in principle there is an obligation on him (the one with eye pains) to not separate from the community, and only now because his eyes hurt him, and therefore after his pain is removed he'll repay the fast; however this does not apply to someone pregnant or breastfeeding, who to begin with at their core are exempt, even if they are not in pain. But from the words of the Bach it seemingly does not indicate this.

Should not separate: Even one traveling on the road and it is difficult for them to fast

You are allowed to eat for the circumcision - That is all the people who were invited for the feast, even more than ten, and even in the morning, they are allowed to eat, but they must all fast the next day as will be noted. And see the Taz and the Elya Rabba that disagree with this gloss and their view is that the fast is fixed for Thursday and therefore they are all forbidden to eat on Thursday and they should have the feast at night. But the Baalei Brit themselves are allowed to eat on Thursday after the Milah and don't need to eat the next day, like it is stated in Siman 559:9 concerning Tisha Ba'av; and see the Pri Megadim who wrote that one who is lenient to follow Rema's opinion [here] has not lost; and similarly in Mor uKetziah and Shevus Yaakov [those scholars] agree with Shulchan Aruch (i.e. this lenient ruling)