UPDATE:
I tried for hours to get through to Airbnb that same day the woman booked (twitter, email, call 3x). Finally that evening at 9:30pm I tried calling the superhost hotline a 4th time, and immediately got through to a rep. Unbelieveably, I spent the next 30 min arguing with the phone rep. I repeately asked if 3rd party bookings were against Air's ToS. Not once did she confirm (or deny). Instead, she repeatedly DEFENDED the guest, saying "the mother was just trying to do something nice for her child, and book a vacation for the child and babysitter."
In addition, I was chastised for "waiting til the night before checkin" not telling the guest MYSELF by message that I had a problem with her booking for a third party, AND that I "don't want kids" (the rep's words, not mine). (uhh...duh...Nevermind that the booking had only originated mere hours earlier.)
[I still think it's very suspicious why a mother would book a place to ship off her kid and a babysitter LESS THAN 24 HRS later]
I gave up in frustration on the phone. The rep said she would immediately call the guest and cancel the booking, but ALSO help her to FIND ANOTHER place instead for the mother to third-party book for the babysitter/child. :0
Half hour later, I get a message from the rep saying that she called and told the guest that I WAS UNCOMFORTABLE HAVING CHILDREN HERE and that I WANTED THE BOOKING CANCELLED. (and that yes, she had found another place for them to book instead.) She totally threw me under the bus!
After I complained again to Airbnb, this is the response I got:
"If they were bringing a child, that is their own choice, but it doesn't have anything to do as far as our ToS goes, but still, this should've been handled better, and much more professional than it was. We are always wanting to make both our Guests and our Hosts feel appreciated and respected by each other, as well as us, and to keep the platform safe and that everyone is treated fair."
--->>> I thought that was the end of it, but now I am getting harrassing messages through the app from the mother saying it was a bait-and-switch, that she was furious, and that she is working to ruin my reputation, and is working with Airbnb to leave a bad review for me for the non-booking. She clearly has also been digging into my background/profile on the internet, because she mentioned other places I am associated with that she was intending to get the word out not to do business with me.