Burying the children of a Kafir in Muslims cemetry
It is not permissible to bury a Kafir in Muslims graveyards whether such a Kafir was adopted by a Muslim or not and regardless of the issue of attaining or not attaining the age of puberty. However, if there is a sign that such a person embraced Islam; they may be buried in the Muslims graveyards. Finally, you must be aware that adoption is Haram (prohibited) in Islam for Allah (Exalted be He) says:
Call them (adopted sons) by (the names of) their fathers
May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.