Gemara

Rav (and R. Yochanan): One is Yotzei with Kidush in Beis ha'Keneses even if he will not eat there;

Shmuel: One is Yotzei only if he will eat there, e.g. visitors who eat in the Beis ha'Keneses. Shmuel would not rely on Kidush on the roof to eat in the house.

Rishonim

Rif and Rosh (10:5): The Halachah follows Shmuel.

Source: Rabah and Rav Huna hold like him. This overrides the general rule that the Halachah follows Rav or R. Yochanan against Shmuel.

Rambam (Hilchos Shabbos 29:8): Kidush must be where one eats. One who made Kidush in one corner may eat in a different corner (of the room). We make Kidush in the Beis ha'Keneses to be Motzi visitors who eat there.

Source (Hagahos Maimoniyos 30): The Rambam learns from the Yerushalmi (Berachos 6:6), which permits to make Kidush in the house (on Shemini Atzeres) and eat in the Sukah.

Magid Mishnah: The visitors hear Kidush and eat in houses next to the Beis ha'Keneses, for one may not eat in a Beis ha'Keneses.

R. Nisim Ga'on (brought in Rosh 10:5): The Yerushalmi (cited in 2:i) allows Kidush in another place if he intended to eat there from the beginning.

Rebuttal (Rosh ibid.): The Bavli disagrees. Alternatively, the Yerushalmi says only that intent helps from one part of the house to another.

Question: According to R. Nisim, why did the Gemara ask why we make Kidush in the Beis ha'Keneses according to Shmuel? One who intends to eat in his house can be Yotzei with it!

Answer Rosh (ibid.) The Gemara thought that this can lead people to think that Kidush need not be b'Makom Seudah. People will not realize that it is for people who intend at the time and want to be Yotzei now, for people normally repeat Kidush at home.

R. Yonah (brought in Rosh ibid.): Mid'Oraisa, Kidush need not be where one eats. Therefore, one may make Kidush in a Beis ha'Keneses even if no one eats there, and he is Yotzei mid'Oraisa; it is not a Berachah l'Vatalah. It is our custom to do so on behalf of people who do not know how to make Kidush for themselves.

Rebuttal (Rosh ibid.): According to R. Yonah, why did the Gemara ask why we make Kidush in the Beis ha'Keneses according to Shmuel? It is for people who do not know how to make Kidush for themselves! Rather, if no one eats there no one is Yotzei, and the Berachah is l'Vatalah.

Poskim

Shulchan Aruch (OC 269:1): The custom is to make Kidush in the Beis ha'Keneses. A child tastes the wine, but the one who made Kidush (or any other adult) should not, for one is Yotzei Kidush only where he eats.

Beis Yosef (DH veha'Mekadesh): We do not feed any Isur forbidden by a Lav to children. However, eating before Kidush is only an Isur Ase. Some forbid feeding even an Isur mid'Rabanan to children. Nevertheless, since one opinion permits an adult to drink it (Rav, who does not require Kidush b'Makom Seudah), one may give it to children to avoid a Berachah l'Vatalah. According to the opinion (brought in Tur 273) that a Revi'is of wine is considered a Seudah, an adult may drink a Revi'is. Our custom is not to do so.

Magen Avraham (1): The Bach requires giving the wine to a child below the age of Chinuch (training in Mitzvos). However, the Rashba calls this a Berachah l'Vatalah; rather, we give it to a child at the age of Chinuch regarding Berachos. Chachamim did not forbid to children things that promote their growth such as food and drink, especially when the food is permitted at another time. One may not pain children by forbidding them to eat before Kidush.

Note: The Magen Avraham himself (551:31) says that during the first nine days of Av, we give Havdalah wine to a child too young to mourn over Yerushalayim. Mishnah Berurah (551:70) agrees, and additionally requires the child to be at the age of Chinuch regarding Berachos (which is less than the age of Chinuch for Aveilus). Here, the Magen Avraham assumes that the age of Chinuch regarding Kidush is not above the age of Chinuch regarding Berachos, therefore he rejects the Bach.

Magen Avraham (ibid.): If there is no child in the Beis ha'Keneses, an adult may drink the wine, with intent to be Yotzei for the meal he will eat in his house. Nevertheless, he may make Kidush again in his house (on behalf of his household - Mishnah Berurah 1).

Shulchan Aruch (ibid.): Kidush in the Beis ha'Keneses was enacted for visitors who eat there. The custom persists in many places even nowadays, when no one eats there. It is better to adopt the custom not to make Kidush, and this is the custom in Eretz Yisrael.

Beis Yosef (DH v'Nohagin): The Ran defends the custom, for it was enacted for visitors. Similarly, the Berachos of me'Ein Sheva (Magen Avos...) and Chazoros ha'Shatz were enacted for particular reasons (lest people return alone from the field at night, and to Motzi people who do not know how to pray), but we say them even when these reasons do not apply!

Magen Avraham (3): The Rashba defends the custom, for Ge'onim enacted it. Ta'anis 24a testifies to its virtue. (A certain man used to get wine to be Motzi a very poor Tzibur in Kidush and Havdalah; his prayer brought rain immediately.)