Where can I purchase all the children's gift items?
You can purchase those gifts from general merchants (Bits n Pieces in Solitude, Lucan Valerius in Riverwood, Gray Pine Goods in Falkreath, etc), under the "misc" section.
Note that children's clothing will appear also under "misc", not as "apparel".
Radiant Raiments has the best child apparel and the hidden chest in Dawnstar has a wooden sword for the boy and a doll for the girl.