Is it permissible for the Aqiqah for children to be offered by the grandfather, or for it to be offered from one cow slaughtered for twins and a nephew together? And is it obligatory to cook the meat of the Aqiqah or to distribute it?

The Aqiqah is a Sunnah, according to the preponderant opinion, for whoever is responsible for the child's maintenance. Your father is permitted to perform the Aqiqah for your children and your brother's child, with the permission of your husband and your brother. Some scholars have held that it is permissible even without their permission. A cow or a camel suffices for seven, by analogy with the Hadi (sacrificial animal for pilgrimage) and the Udhiyah (sacrificial animal for Eid al-Adha). Therefore, slaughtering one cow is sufficient for your children and your brother's child, provided that the cow has completed two years of age and is free from defects that prevent its validity. The Aqiqah should be dealt with in the same way as the Udhiyah, regarding eating, gifting, and giving in charity.