Do children have to keep the minhag of not eating meat in the nine days? If yes, from what age? In addition does this apply even if they do not like milky products?
It is permitted to give meat to children who are beneath the age of chinuch for mourning over the Mikdash. According to most authorities, this includes children up to the age of 9-10. This applies even when the child does not like milky products, and there are various alternatives to meat that can be used (soya-based or fish-based products, and the like).
Sources: Magen Avraham, 551:31; Chayei Adam 133:16; Or Le’zion III, 26:6; concerning the age of chinuch, see opinions cited in Piskei Teshuvos, Vol. VI, 551:35, p. 180.