Is it permissible to slaughter the Aqiqah for children in Ramadan with the intention of feeding the fasting person, and is it permissible to offer it to other than the eldest child, and must the Aqiqah be a sheep?

Aqiqah is a confirmed Sunnah. Its preferred time is on the seventh day. If that is missed, then on the fourteenth, or the twenty-first. If one is unable to do it at these times, then he may perform the Aqiqah thereafter whenever his circumstances permit, and there is no harm in slaughtering it in Ramadan with the intention of feeding those who are fasting. There is no requirement for a specific order among the children. It is sunnah to sacrifice two sheep for a boy and one sheep for a girl, though one sheep is sufficient for a boy.