Is the 'aqiqah valid if the head, hide, and intestines were left for the butcher who slaughtered and sold it, with full payment given to him? And must it be redone? And is it better to slaughter two goats for a male or one sheep?

If you give a part of the 'aqiqah to the butcher as a gift, there is no harm in that. The ruling on 'aqiqah is the same as that of the udhiyah (sacrificial animal), and it is permissible to give some of it to the butcher as a gift. As for the butcher selling what was gifted to him, there is no harm in that.

However, if you slaughter two goat kids for a male, this is not sufficient, unless what is meant is a male goat that has passed one year of age. It is better for a male to have two sheep, based on the Prophet's (peace be upon him) saying: "For a boy, two sheep." The term 'sheep' (شاة) applies to both lamb (ḍa’n) and goat (ma’z).

Therefore, performing the 'aqiqah for a boy with two sufficient goats is better than with one lamb, even though the species of lamb is superior.