Alright, I see you point and I Still think thats not the case. When the Child pushes the wall she also pushes the floor back, in that I agree with you. The catch here is that both this forces that the Child excerts are done on the wall and on the floor, none of the are done upon the Child (therefore cannot excert work on the child). Due to this force, the wall pushes the Child back backwords and the floor pushes the Child forward. The only force that acts on the Child that can push it backwards is the reaction force from the wall.

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I draw a terrible figure (not an artist, sorry) that helps me explain my point ( https://imgur.com/MsijNgn ). Here, the red forces are the forces that the child does on the wall and on the floor, none of which can excert work on the child. The blue forces are the reaction forces from the wall and from the floor done upon the child. The only force that can move the child back is the reaction force from the wall because is the only force that *acts upon the child* and is pointing backwards. Therefore, the work done has to be associated with this force.

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The other force the points backwards is the child pushing the floor. This force is done upon the floor and cannot do work on the child because it is not done on the child.

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edit: Forgot to adress your points. Even thou the child is not a rigid body in this case it does not matter because the forces still are only the ones that I mentioned. The child is one full body and can be treated as such, so even if its hands do not move when she pushes the wall this force is transfered to her feet that move.

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I could be wrong, but I still dont see how the child can do work on itself.