SEIF 1
BISHUL AKUM: EXCEPTIONS

The Rabanan prohibited food that was cooked by a Goy. The Gemara in Avodah Zara 38a talks about the exceptions as paraphrased by Kollel Iyun Hadaf
(a) Version #1 (in Sura - Rav Shmuel bar Rav Yitzchak): Anything that can be eaten raw, Bishul Akum does not apply to it.
(b) Version #2 (in Pumbadisa - Rav Shmuel bar Rav Yitzchak): Bishul Akum does not apply to anything not fit for a king's table.
(c) Question: What is the difference between these answers?
(d) Answer: Small fish, mushrooms, and porridge (Rashi- they are not eaten raw but they are not fit for a king's table; Rashba - small fish and mushrooms are not eaten raw but are not fit for a king's table, porridge can be eaten raw but is fit for a king's table).
According to Rashi's explanation it only lists how fit for a king is more lenient than eaten raw. It doesn't list anything that fit for a king would be more stringent than eaten raw. From this most Rishonim conclude that the later version is just adding a leniency to the first version and not outright arguing with it. Since we Paskin like the later version, thus we can permit anything that either not fit for a king's table or anything that is edible raw.

SEIF 1 CONT.
THE REASON FOR BISHUL AKUM

Rashi says the reason for Bishul Akum, that if you get used to eating food cooked by Goyim, even if those foods where watched and made sure there was nothing not Kosher in it, you'll come to eat one that was mixed with non-kosher ingredients.

The Rambam says that its prohibited because you don't want to socialize with them so you won't come to intermarriage. The reason why its permitted if its not fit for a king, since over such foods people don't socialize.

The PT in the name of the Teferes Moshe says that the Nafka Mina between the two reasons would be eating something that was cooked by a Jew that isn't Frum. There is no problem of intermarriage, since they're Jews, but you have a problem about mixing in non-Kosher ingredient. (I heard from a big Posek that you can be Maikel and allow food cooked by someone who isn't Frum)

COOKING IN A JEW'S HOUSE.

Rabeinu Avraham wanted to allow a Goy to cook in the Jew's house, since according to both reasons, it wouldn't apply. Since its in the Jew's home, there aren't any non-kosher ingredient to mix in and since its all his food, so you its not going to lead to socializing with a Goyisha host.

Rabeinu Tam says its still forbidden, the Rabanan prohibited in all cases, even when technically the reasons didn't apply, as a "Lo Plug." (To prohibit across the board and not to make exceptions.)

AN IMPORTANT PERSON (ADAM CHOSHUV, A TZADIK)

The Gemara in Shabbos 51a says (and is paraphrased by Kollel Iyun Hadaf)

(k) On Shabbos, Rav Nachman told his slave to Matmin (wrap up, usually to keep hot, here to keep cold) water; on a weekday, he told his slave to bring hot water from a Nochri. R. Ami was upset.
(l) Question (Rav Yosef): R. Ami should not have been upset - Rav Nachman followed his Rebbi'im, Rav and Shmuel!
1. (Shmuel): One may Matmin something cold.
2. (Rav): Bishul Akum (the Isur mid'Rabanan to eat something cooked by a Nochri) does not apply to something that can be eaten raw.
(m) Answer: R. Ami holds that an important person should be stringent. From here the Shach in 152:2 says that an important person should be stringent and not eat something that was cooked by a Goy even though it could have been eaten raw. The Dagal Mervava disagrees, that we should Paskin like Rav Nachman over Rav Ami, since he did an action (Maaseh Rav) that showed he relied heavily on his P'sak, so the Halacha is like him that even an important person can be lenient. He also quotes the Magen Avraham 257:15 that says by wrapping up cold things on Shabbos that even an important person is allowed to do it.

SEIF 2
MIXTURES

If there is a mixture of ingredients, some that have a problem of Bishul Akum, and some that doesn't (not fit for a king or eaten raw) then the status of the food depends on the main ingredient. If its something that has a problem of Bishul Akum, then the whole mixture is forbidden, if not, then its permitted. This is learnt from the Gemara in Avoda Zara 38a (and is paraphrased by Kollel Iyun Hadaf).

1. If a Nochri made Kisa d'Harsena from them (fried them in fish oil with flour), they are forbidden on account of Bishul Akum.
2. Objection: This is obvious!
3. Answer: One might have thought, the primary ingredient is the oil (and this can be eaten raw) -Rav Yosef teaches, the primary ingredient is the flour (which is not eaten raw). Thus the status goes after the main ingredient.

CONTINUE OF FOODS THAT CAN BE EATEN RAW OR FIT FOR A KING

The Gemara in AZ 59a says
(g) R. Chiya bar Aba went to Gavla and saw three things etc. Yisraelim would eat lupines cooked by Nochrim.
1. He did not say anything to them; he told R. Yochanan, who told him to announce that and their lupines are forbidden as Bishul Akum because they are not Benei Torah. The Rabeinu Yeruchim brings a Machlokes Rishonim if nowadays are we considered in general as not B'nai Torah as much as the people of Gavla and nor permit any food cooked by a Goy or that we're better than them and are able to have all the Heterim written by Chazal. He concludes that the main opinion is that its permitted and that is how we Paskin. We also do not have to worry (in a case where the standard procedure there is no non Kosher ingredients added) that it was cooked in a Goy's pot and then forbid the food. Since we have a rule that an average pot wasn't used within 24 hours and the absorptions are stale and doesn't give good taste into the food and its permitted.

SEIF 3
MEAT OR FISH PIES

If a Goy baked meat or fish pies, even for those that are not Makpid on Pas Yisrael, it is prohibited. Since the meat or fish falls under Bishul, there is no Heter for it and is prohibited. Its not compared to the eggs that where mixed into the dough of bread, because that was only an ingredient which was Batul to the main ingredient, so we
look at the dough as taking the status of its main ingredient, which isthe bread.

But over here, since the meat and fish isn't mixed into the dough, but its a separate entity, so it gets its own status. Even if one would scrape off the meat/fish the crust would be forbidden. Since the fat of the meat/fish, which originally became a prohibition by being cooked in its own entity, is absorbed into it, so in
essence you are eating the Issur. Even though we said (when we
discusses Pas Akum) that these types of Issur D'rabanan are Batul B'Rov (with just having the majority Heter) over here, since you can recognize the globs of fat in the crust, it's considered as the Issur is recognizable and thus is not Butul.

SEIF 4
NON JEWISH MAIDS

The Ramban says that maids that were bought (but where not made into Shipchos Kannanos) if they cook for you, its permitted. His main reasoning, since they are owned by you to the point that you're obligated that they cannot work for you on Shabbos (its included in what the Torah says "your servant and maidservant" should rest.) thus they go out of the category of a regular gentile. The Ra'ah and Rashba disagree. The Rama says the we can permit it.

The commentaries agree that the original Heter doesn't apply nowadays, since we don't own our maids (but rather they're hired hands) and are not obligated to have them rest on Shabbos. But they permit it based on three principles. 1) the Rabeinu Avraham we had in the first Seif that permits Bishul Akum if its done in the Yisrael's house. Though we don't Paskin like that, but with the addition that 2) a Svara that Bishul Akum only applies when the goy is doing it at his own volition, because that's more a social event that might lead to intermarriage, and not what's part of their jobs which they are forced to do.

But mainly because of this third reason, that since its done at the Jew's house (according to the Ashkenazim) its most probable that a Jew (one of the household) had stroke the coals (strong enough to help the cooking) that would permit the food. (This is proof that you don't need to have in mind with the stroking of the coals to permit the food.) So even if we don't know for sure if someone stroked the coals, we can assume it.

Would this first reason have any ramification for either the specific animal (an ox I think) who had been owned by a rabbi in the time of the gemara who when sold refused to work on Shabbos for his new gentile owner?

Are there any modern cases of an animal that learned to recognize that there is a difference between Shabbos and the work week?

There was a cute story in recent years about a lost parrot who's first words after resuccitation were "Boruch Hashem"!

Quote "Would this first reason have any ramification for either the specific animal (an ox I think) who had been owned by a rabbi in the time of the gemara who when sold refused to work on Shabbos for his new gentile owner?"

Well, a servant and an animal is supposed to rest on Shabbos is learnt from the same Pasuk.