Is it valid for a child to perform the Sunnah of Ihram and [prayer at] Maqam Ibrahim, and is it sufficient for the Tawaf and Sa'i to be performed with the child being carried or in a stroller, or must the Tawaf be performed by the child independently?

The two rak'ahs of ihram are a Sunnah, not obligatory. It is permissible for the traveler to pray them on the airplane.

Circumambulating (Tawaf) and performing the ritual walking (Sa'i) with your child carried or in a stroller is sufficient for both of you. The majority of scholars hold the view that a guardian does not pray on behalf of a child.