Why can't you flatten a sphere? It's a well-known fact that you can't flatten a sphere without tearing or deforming it.  How can I explain why this is so to a 10 year old?

As soon as an explanation starts using terms like "Gaussian curvature", it's going too far for the audience in question.

A great explanation would work just as well for a hemisphere as for a sphere, as both have positive curvature.
Experimentation is fun!

Give him a globe, some string, a ruler, a compass, some paper, and a pencil.  Then have him try to map 4 different cities from the globe onto the paper, keeping ALL the distances between them the same as on the globe.
You can, just imagine a perfect sphere, being put into a piece of paper that breaks away as it enters, then take a picture of it. Perfect sphere, the reason you can't physically flatten a sphere, is because all of that third dimension of space has to go SOMEWHERE, so it's either going to fall apart, and split the third dimension perfectly (EXTREMELY unlikely) or, it'll stay together, and become deformed.

A perfect sphere is merely a circle anyway, it's just a filled in surface of the spheres diameter.

This is a complete shot in the dark, and I hope my answer helps, but it's obvious so it probably won't :/