Gemara

(Mishnah): A man receives Metzi'os (Hefker objects) that his daughter finds, and her earnings.

Kesuvos 47a: He receive her Metzi'os to avoid enmity.

Bava Metzi'a 12a (Mishnah): Reuven receives Metzi'os found by his small sons and daughters. If his big sons or daughters, or his male or female Yisrael slaves found a Metzi'ah, they keep it.

(Shmuel): Chachamim said that Reuven receives Metzi'os found by his minor son because when he finds it, he immediately brings it to his father. (He picked it up with intention to give it to him.)

12b (R. Yochanan): 'Small' and 'big' in the Mishnah do not refer to minor and adult. Rather, any child fed by his father is called small. Any child who feeds himself is called big.

Question: What is the case of the female Yisraelis slave in the Mishnah?

If she is still a minor, and she has a father, he gets her Metzi'os!

Eruvin 79b (Mishnah): One may ask his big son or daughter to be Mezakeh (acquire) an Eruv on behalf of others. He cannot be Mezakeh through his small son or daughter, because they can acquire only for him.

Rishonim

Rif and Rosh (Bava Metzi'a 5b and 1:33): 'Small' and 'big' in the Mishnah do not refer to minor and adult. Rather, any child fed by his father is called small. Any child who feeds himself is called big.

Rosh: The same applies to an Eruv. One may not be Mezakeh through an adult child if he feeds him. This applies only to a child. If Reuven feeds a stranger, the stranger's Yad (power of acquisition) is not like Reuven's.

Nimukei Yosef (DH Gemara): The Rashba and Ran say that regarding Eruv, 'small' literally refers to a minor. There is no concern for enmity when one is Mezakeh to others through his son. R. Tam disagrees. Some say that just like a father receives Metzi'os of his Katan, he receives a gift given to him. If the father gave a gift to him, the son did not acquire, for he has no Yad. The Ran says that presumably, this applies to a minor fed by his father. An adult is no worse than a wife. She acquires a gift that he gives her, even though her Yad is like his.

Teshuvas Betzalel Ashkenazi (35): Also the Ran says that a Gadol fed by his father does not acquire even from his father, for he has no Yad of his own. However, a woman fed by her husband can be Mezakeh an Eruv! The Ritva and Me'iri answer that since he is obligated to feed her, it is as if she feeds herself. This also explains why she keeps a gift that he gives to her, even though her hand is like his. The Rashba holds that an adult can be Mezakeh an Eruv even if his father feeds him, since there is no concern for enmity. Likewise, he can receive his own gift.

Rambam (Hilchos Aveidah 17:13): A man receives Metzi'os found by his son or daughter whom he feeds, even if he or she is an adult. He receives Metzi'os of his daughter when she is a Na'arah. He does not receive Metzi'os found by his son who feeds himself, even if he is a minor.

Ran (Kesuvos 17a DH Masnisin): Rashi says that a man gets his Na'arah's Metzi'os due to enmity, lest he not feed her. This is wrong. In Bava Metzi'a, we say that a man gets his son's Metzi'os only if he feeds him, but he gets his daughter's even if he does not feed her! I.e. we said that an Amah Ivriyah's Metzi'os go to her father, even though he does not feed her (her master feeds her)! Rather, we are concerned for enmity lest he marry her to a Menuval (a repulsive man, who would pay much to be Mekadesh her), like Tosfos says.

Poskim

Shulchan Aruch (CM 270:2): A man receives Metzi'os found by his son or daughter that he feeds, even if they are adults.

Beis Yosef (DH Metzi'as): It is known that the Halachah follows R. Yochanan against Shmuel. The father receives Metzi'os of children that he feeds, due to enmity. Tosfos (DH R. Yochanan) says that this applies only to children, for it is normal to feed one's child, and if he does not, others will not feed him. If Reuven feeds a stranger, there is no reason why the stranger's Metzi'os should go to Reuven.

SMA (2): Chachamim deemed it proper to repay one who does good. Since he feeds his children, even though he is not obligated after they are six, we enacted that he gets their Metzi'os. Brothers are obligated to feed their sisters after their father died, therefore the sisters keep their Metzi'os.

Taz (DH ha'Semuchim): If one committed himself to feed his child, the child keeps his Metzi'os, for the father is obligated. This is like sisters, who keep their Metzi'os because their brothers are obligated to feed them.

Shulchan Aruch (ibid.): He receives Metzi'os of his daughter when she is a Na'arah, even if he does not feed her.

Beis Yosef (DH u'Mah she'Chosav): Rashi explains that if she would keep her Metzi'os, perhaps he would cease to feed her. The Ri says that we are concerned lest he be Mekadesh her to a Menuval. A man gets his son's Metzi'os only if he feeds him. Regarding a Na'arah, in any case he gets her Metzi'os due to this enmity. The Rambam agrees.

SMA (3): All the more so he gets Metzi'os of his minor daughter even without feeding her, for he can be Mekadesh her to a Menuval, or sell her for a slave.

Shulchan Aruch (ibid.): He does not get Metzi'os found by his son whom he does not feed, even if he is a minor.

SMA (6): A daughter who is not in his Reshus, e.g. she was widowed or divorced from Nisu'in, is like a son.

Rema: If one give a gift to a Katan fed by his father, it is the father's. This does not apply to a Gadol son.

SMA (8): Perhaps one who gave to a Gadol was particular that the Gadol keep it. One who gives to a Katan knows that it will be guarded only if the father guards it.

Question (Gra 6): Also a Katan acquires when someone else gives to him!

Rema: An orphan fed by someone else keeps his Metzi'os.

SMA (9): If the one feeding him would not do so, many others would, for it is a great Mitzvah. Therefore, Chachamim saw no need to enact here.

Shulchan Aruch (OC 366:10): One cannot be Mezakeh food for an Eruv through his minor children, even if he does not feed them, or through his Kena'ani slaves.

Gra (DH Afilu Im): Shmuel agrees with R. Yochanan regarding Eruv.

Shulchan Aruch (ibid.): He can be Mezakeh through his adult children, even if he feeds them.

Mishnah Berurah (57): Even though their Metzi'os go to their father, this is only an enactment to avoid enmity. They have their own Yad. If their father wants them to be Mezakeh for others, it works.