Gemara

Mishnah: A child who knows how to shake a Lulav is obligated in the Mitzvah.

Beraisa: A child who knows how to shake a Lulav is obligated. A child that knows how to cover himself (with a Talis) is obligated in Tzitzis. When a child knows how to guard (Kedushas) Tefilin (not to sleep, emit gas or enter a bathroom while wearing them), his father buys Tefilin for him.

46b - R. Zeira: One should not be Makneh (transfer ownership of) his Lulav to a child on the first day, because a child acquires, but he cannot be Makneh.

Gitin 59a - Mishnah: The purchase or sale of Metaltelim (movable objects) by Pa'utos (mature children) is valid.

Rav Yehudah said that children are considered Pa'utos at the age of six or seven, Rav Kahana said that the age is seven or eight, and a Beraisa says that it is nine or ten.

They do not argue. It varies, according to the sharpness of the child.

Question: Why was it enacted that Pa'utos can buy and sell in Metaltelim?

Answer (Rav Aba bar Yakov): This was so he can buy his necessities.

Question (Abaye): Can Pa'utos give gifts?

Answer #1 (Rav Yemar): They cannot.

Answer #2 (Mar bar Rav Ashi and Rav Mordechai in the name of Rav Ashi): They can. This applies whether the child is healthy or about to die, whether the gift is large or small.

64b - Rav Yehudah: If a child knows to discard a rock but keep a nut, he can acquire for himself, but not for others. If the child can take a deposit and return it later, he can acquire for himself and for others.

Shmuel: Both of these children have the same law.

Rav Chisda: He means that either can acquire for himself, but not for others.

Question (Rav Chinena Vardan - Mishnah): The following may acquire Shituf Mavo'os (a kind of Eruv) - ...a female Yisrael slave.

A female Yisrael slave must be a minor, yet she can acquire for others!

Answer: Shituf Mavo'os is mid'Rabanan. Chachamim were lenient to allow her to acquire, but mid'Oraisa, she cannot acquire.

Rav Chisda: Rav Chinena could have countered that decrees of Chachamim are patterned on mid'Oraisa laws.

Rishonim

Rif and Rosh (Gitin 27b and 5:19): Chachamim enacted that Pa'utos can buy and sell for their needs. If they have a (parent or) overseer, they cannot.

Ran (DH Garsinan): R. Zeira discusses a child below Pa'utos. Pa'utos can buy and sell Metaltelim, and the conclusion (Gitin 59a) is that they can give gifts. Regarding a child below this, we do not say that since he did not return it (properly), the gift was not a gift. Rather, a Tanai (condition) that is impossible to fulfill is not a Tanai, so it was an unconditional gift.

Rambam (Hilchos Mechirah 29:1): Mid'Oraisa, a child cannot buy or sell. Chachamim enacted that he can buy and sell Metaltelim Kedai Chayav.

Rambam (6): Until six years he cannot be Makneh anything to others. From six until adulthood we check him. If he understands buying and selling, his purchase, sale and gift are valid, for big or small matters.

Rambam (Hilchos Zechiyah u'Matanah 4:7): If a child knows to discard a rock but keep a nut, he can acquire for himself, but not for others. Before this he cannot acquire even for himself.

Rambam (Hilchos Lulav 8:10): We do not give a Lulav to a child on the first day of Sukos.

Question (Kesef Mishneh): The Ran says that this is only if the child is below Pa'utos, i.e. six or seven. Why didn't the Poskim mention this?

Answer (Lechem Mishneh and Magid Mishnah, Mechirah 29:1): The Rambam holds that Pa'utos acquire mid'Oraisa when an adult gives to them, but they cannot be Makneh mid'Oraisa.

R. Mano'ach: A child that knows why and how to shake is surely old enough to acquire for himself. If he reached Pa'utos, just like he acquires mid'Rabanan he can return mid'Rabanan. An adult should not acquire on behalf of any minor, for this is mid'Oraisa and there is no way to return it mid'Oraisa.

Rosh (4:6): One should not be Makneh his Lulav to a child on the first day, because a child acquires, but he cannot be Makneh. The giver will be unable to be Yotzei.

Hagahos Ashri: A child is unlike a lunatic, who cannot acquire at all. If one holds it with the child, since it never left his hand this is fine. Wise people put the Lulav away and let the child take it and bless on it.

Rosh (3:30) If a giver said 'this is a gift until you are Yotzei, and then it reverts to me', the recipient is not Yotzei, for this is like a loan. If a limited time gift was valid, one could give to a child and get it back, even though the child cannot give!

Rebuttal (Ketzos ha'Choshen 241:4): R. Shimon ben Gamliel (Bava Basra 137) holds that when one says 'I give you this to you, and after you to Ploni', he gives Kinyan ha'Guf, and even Rebbi agrees regarding an Esrog (if not, why did he give it?!) A limited time gift is just like 'after you, to me', just it does not allow the recipient to eat or sell it. Tosfos (Erchin 30) says that Chalifin does not work for things that return, such as a gift on condition to return it. If it is a permanent Kinyan on condition, surely Chalifin should work! Rather, it is a limited time gift. The Rosh (Teshuvah 35:2) cites R. Avigdor, who says that it suffices to give a Lulav for a temporary Kinyan ha'Guf. R. Zeira teaches that one should not it give his Lulav for a permanent Kinyan to a child.

Poskim

Shulchan Aruch (658:6): One should not give his Lulav to a child on the first day before the giver was Yotzei, because a child acquires, but he cannot be Makneh (back) mid'Oraisa. Even if he returns it, it is not returned. Some permit if he reached the age of Pa'utos.

Bi'ur Halachah (DH Lo): The latter opinion holds that even when an adult gives to them, children acquire only mid'Rabanan. Pa'utos can be Makneh back mid'Rabanan. The first opinion holds that Pa'utos acquire mid'Oraisa when an adult gives to them, but they cannot be Makneh mid'Oraisa,

Eshel Avraham (8): In Chutz la'Aretz, one is not Yotzei with a borrowed Lulav on the second day. Therefore, one cannot give his Lulav even after he was Yotzei on the first day. To fulfill Chinuch one must buy a Lulav for the child himself

Note: Tosfos (Erchin 2b DH Aviv) says that the Beraisa mentions buying Tefilin for one's son, but it does not say that he buys a Lulav for him, because the son can be Yotzei with the father's. Tosfos could hold like R. Avigdor, that a temporary Kinyan ha'Guf suffices, or he could hold that the child need not acquire it to fulfill Chinuch (see Bi'ur Halachah below). Alternatively, the Tana did not say that he must buy a Lulav for the child, for it is not always necessary. In Eretz Yisrael (or in Chutz la'Aretz, if one of the first two days is Shabbos) the father can be Makneh his after he was Yotzei. According to the latter opinion in the Shulchan Aruch, there is never a problem if the child is above Pa'utos,

Kaf ha'Chayim (56): The Levush says that one should not give it even after he was Yotzei, lest he need to give it to another adult. Likewise, in Chutz la'Aretz one should not give it to a child on Yom Tov Sheni even after he was Yotzei. However, one who normally does not give his Lulav to adults need not be stringent about this.

Kaf ha'Chayim (62): The Eshel Avraham says that one should be stringent about a 13 year old if we are unsure whether or not he brought two hairs, for it is a Safek mid'Oraisa. The Kesav Sofer is lenient because it is a Sefek-Sefeka; perhaps even Pa'utos can be Makneh back.

Kaf ha'Chayim (65): One should not use a Lulav of the poor or the Rabim on the first day because children have a share in it.

Kaf ha'Chayim (66): The Chasam Sofer says that one should not buy the four Minim from children. If one used such Minim on the first day he should take others, but without a Berachah.

Shulchan Aruch (ibid.): If one holds it with the child, since it never left his hand this is fine.

Source (Gra DH v'Im): In Bava Metzi'a (7a) it says that if Reuven holds a garment and Shimon takes hold of three by three fingers of it, he acquires.

Bi'ur Halachah (28): This is even for children below Pa'utos, for the child does not acquire at all. The same applies if one is not Makneh to the child at all, or if he says 'it is yours until you are Yotzei, then it reverts to be mine.' However, these help only the adult to be Yotzei afterwards, but the child is not Yotzei and Chinuch was not fulfilled. Some say that one fulfills Chinuch even through borrowed Minim, for the child becomes habituated to Mitzvos.