Are children obligated in Havdala?
A child who has reached the age of understanding the concept of Shabbos is obligated to hear Havdala. [This is at approximately 5-6 years old.]
Repeating Havdala for a child: One may repeat Havdala to fulfill the obligation of [male] children [that have reached the age of education and do not know to say Havdala themselves]. [One also the blessings of Besamim and Meorei Haeish on their behalf.] Based on this even if one heard Havdala in Shul he may still say Havdala for his younger [male] children upon his return home.
Hearing Havdala from a child? An adult is not Yotzei if he hears Havdala from a child under Bar Mitzvah. Other children however do fulfill their obligation. Furthermore even if the child is Bar Mitzvah, if he does not yet have a beard, one may not be Yotzei with him unless one knows for certain that the child has already grown two pubic hairs.
If the child did not hear Havdala on Motzei Shabbos is he to hear Havdala on Sunday?
If the children will be sleeping by the time Havdala is said what is one to do?
Some write that in such a case one may be lenient to have them say Havdala after Plag Hamincha on Shabbos.
If a boy turned Bar Mitzvah on Motzei Shabbos may he say Havdala for others that Motzei Shabbos?
It is best to not have others be Yotzei with his Havdala.
May children eat prior to Havdala?
Yes. It is forbidden to refrain them from eating if they need to eat.
 SSH”K 58/19
 See the previous Halacha
 This applies even if they are not members of his family, and their obligation of education is hence not upon him. [167/23; See Kitzur Halachos Shabbos p. 114]
 Ketzos Hashulchan 96/5
 269/3; Kitzur Halachos 296 footnote 32
 One is not required to smell the Besamim when repeating it for children which have reached the age of Chinuch as explained in 297/7 regarding one who cannot smell.
 Ketzos Hashulchan 99/7; Kitzur Halachos 298/16 based on 178/23 and 297/7 that one say all the blessings of Birchas Hanehnin for a child in order to educate them in Mitzvos. Seemingly the same would apply here, even though it is viewed as Birchas Hamitzvos and not Birchas Hanehnin.
 Based on Admur 271/7; See Piskeiy Teshuvos 296/19-20 regarding women being Yotzei with children.
 SSH”K 58/20
 See Piskeiy Teshuvos 296 footnote 182
 Piskeiy Teshuvos 296/19
 Piskeiy Teshuvos 196/19 based on Ketzos Hashulchan 96 footnote 12
 As it is questionable whether he is fully obligated in Havdala being that on Shabbos he was not yet Bar Mitzvah and was thus not yet obligated to keep Shabbos. [Ketzos Hashulchan ibid]
 269/3 regarding Kiddush