Rambam - 1 Chapter a Day

And is governed by the laws stated in Halachah 3. Similarly, the animal taken by the priest cannot be offered as a firstborn offering, because it is possible that it is an ordinary animal and thus one would be slaughtering an ordinary animal in the Temple Courtyard.
The rationale is that it is equally possible that either of the animals is the firstborn. One must be given to a priest, but he cannot take possession of the more choice animal, because we follow the principle: When one (the priest) desires to expropriate property (the more choice animal) from a colleague (the owner), the burden of proof is on him. Since the priest cannot prove that the more choice animal emerged first, he may not take possession of it. On the other hand, the owner cannot consider it as a totally ordinary animal either, for perhaps it is the firstborn.

This is a general principle applicable in many contexts of Jewish Law; see Hilchot Terumot 10:14, Hilchot Mechirah 20:5, et al.

Since the priest cannot prove that the living animal was born before the dead one, he may not take possession of it.

And is governed by the laws mentioned in Halachah 3. Parallel concepts that apply to firstling donkeys with regard to this and the situations mentioned subsequently are found in the later halachot of Hilchot Bikkurim, ch. 12.

And if the first issue of an animal’s womb is female, it is not consecrated.

Even though it is not known which mother gave birth to which offspring, it is of no consequence, for both must be given to the priests.

For the male is definitely a firstborn.

I.e., one animal gave birth to a male and the other to a male and a female and it was not known which - the male or the female - was born first. Thus that male is a firstborn of doubtful status. The two males then became intermingled and it was not known which one was definitely a firstborn and which was of doubtful status.

Since it is not known which is definitely a firstborn, but there is definitely a firstborn there, the priest must be given one animal, but the owner has the choice which animal to give him.

And in both instances, it is not known which one, the male or the female, was born first.

And the priest cannot prove it is a firstborn.

Since it is not known whether the sheep which did not give birth previously was the one which gave birth to the male or not, its status cannot be established definitively.

Once it becomes blemished, it may be eaten by its owners, because there is no sacred quality connected with it. It is not given to a priest because the priest does not have a claim that he can substantiate which will entitle him to expropriate the firstborn from the owner.

As mentioned in the notes to Chapter 2, Halachah 6, the same principle that has been used until now in support of the owner can now be cited in support of the priest. It is not certain that it is not a firstborn and it is now in the possession of the priest. Therefore if someone (in this instance, the owner) desires to expropriate property (the firstborn) from another person (the priest), the burden of proof is on the former. Since he cannot prove that it is not a firstborn, the priest is allowed to retain possession.
Many have questioned the Rambam’s decision. Indeed, volumes have been written about this issue. With regard to practice, there is a difference of opinion regarding the matter between Sephardic and Ashkenazic authorities. The Sephardic authorities (see Shulchan Aruch, Yoreh De’ah 315:1) follow the Rambam’s view and allow the priest to maintain possession. The Ashkenazic authorities (Tur, Rama, lac. cit.) follow the opinion of Rabbenu Asher who states that since the priest had no right to take the animal, it is expropriated from his possession.

Nor is it permitted for him to shear it or work with it (see Hilchot Me’ilah 1:8).

Which is forbidden, as stated in Hilchot Shechitah 2:1.

In which instance, were each to be giving suck to its own offspring, the priest would be given the male animals.

And a distinction was not - or could not be - made to whom which animal belonged.

He is not held responsible. In Hilchot Sha’aleh UPikadon 5:4, the Rambam states that this law applies only when the two individuals entrusted the animals to the shepherd without him knowing. Otherwise, he is required to make restitution to each one.

I.e., after it becomes blemished.

Thus the owner is allowed to retain possession. For the one who entrusted the animal must prove that the living animal is his and he cannot.

Because it is possible it is the firstborn that was entrusted for safekeeping.

In contrast to the priests. See Chapter 2, Halachah 17.

I.e., the Israelite is saying that the animal is a firstborn of doubtful status and it was blemished in his possession on its own accord (Shulchan Aruch, Yoreh De’ah 314:10). Were a priest to make such a statement, we would not accept his word, because we would suspect that he intentionally caused the animal to become blemished.

Since it had a blemish, we understand that his intent in consecrating it was to sell it and purchase a sacrifice with the proceeds of the sale.

Since there was never any possibility that the animals be offered as a sacrifice, the consecration never affected their actual physical bodies. Hence, after they are redeemed, they are considered entirely as ordinary animals.

Since at the time they were consecrated, there was a possibility for them to be offered as sacrifices, the consecration affects their actual physical bodies. Hence, even after they are redeemed, they are never considered entirely as ordinary animals.

And since a vestige of their consecrated quality remains, there is no obligation to set aside their firstborn.

Hilchot Me’ilah 1:9.

I.e., he redeemed the second tithes and brought that money to Jerusalem to purchase food.

The second tithe is considered as belonging to “the Most High” (Hilchot Ma’aser Sheni 3:17) and thus there is reason to think that the offspring would be exempt from the requirements of the firstborn, as the offspring of consecrated property is (see Chapter 1, Halachah 5). Indeed, for that reason, a person who steals such an animal is not required to make double restitution, nor may one use it to consecrate a woman (Hilchot Ishut 5:4). Nevertheless, since the animal was purchased for the intent of food, it is considered as the person’s private property and there is an obligation to redeem its firstborn (Rav Yosef Corcus).

For the produce of the Sabbatical year may be exchanged for other foodstuffs.

See Hilchot Shemitah VeYovel, ch. 6; see also Hilchot Sh’luchin VeShutafim 5:10.

The Ra’avad notes that the standard published text of Bechorot 12b states ‘“to partake of it,’ and not to burn it.” Thus were the offspring to be obligated to be offered as a firstborn, the fats and organs would have to be burnt on the altar’s pyre, violating the above restriction.
Rav Yosef Corcus explains that the proof mentioned by the Ra’avad is debated by the Talmud and requires much effort to substantiate, while the rationale stated by the Rambam is more straightforward. Hence he employs it even though it is not stated in the Talmud.

I.e., there is a specific time when the produce of the Sabbatical year has to be destroyed and, by exchanging the produce for the animal, the person releases himself from that obligation. Moreover, the priest to whom he gives the firstborn will be grateful to him. Since the purchaser derives these benefits from the exchange, he is considered as “performing commercial activity.”

Hilchot Ma’achalot Assurot 8:16-18.

Hilchot Terumot 12:21.

I.e., to buy and sell with the intent of making a profit (Shulchan Aruch, Yoreh De’ah 307:3).

Chapter 1, Halachot 17-18.

I.e., a blemished firstborn, for an unblemished firstborn could not be slaughtered for such a purpose.

For he did not purchase it with the intent of making a profit.

The term “evaluate an animal” refers to an arrangement in which an animal’s present worth is evaluated. It is then given to a shepherd who will care for it and feed it for a specific time. Afterwards, it will be sold and the profits divided between the owner and the shepherd [the Rambam’s Commentary to the Mishnah (Terumot 11:9)]. Since an Israelite does not have a right to an unblemished firstborn, such an arrangement cannot be made with him, for he may not take appropriate care regarding the matter (Rav Yosef Corcus).

But not in the era of the Temple, for in that era, the priests do not have any rights to a firstborn, for it must be offered as a sacrifice and may not be used for personal benefit.