My suggestion would be to get the passport quickly through the USA Passport Agency near you (which tends to be quicker than applying by mail), but you don't need to pay the extra fees for same day service (unless you live far from them and want the convenience of only making one trip, although IIRC they can mail the passport to you very promptly). BTW, I'm not sure that needing the passport number to get plane tickets is an acceptable reason for very fast service, or even how you would document this, so just booking a refundable dummy international ticket might be the easiest way. (For a child traveler, do you need to show tickets for an adult too? If so, this could be a good time to obtain/renew your own passport.)

I'd also book the kid's plane ticket very soon, before seat availability disappears and before prices rise.

Don't mess around with trying to change the name on an existing ticket. Most airlines allow this only for minor name corrections and not to change the identity of the person who will be using the ticket to travel.

Don't worry about having child and grandparents on the same reservation (PNR). You can call the airline to tell them to "link" the tickets and advise them that these adults are accompanying the child.

In this case, it's clearly worth paying seat fees to get seats together now rather than worrying about this on the day of departure.