Duality

Would like to ask if these are, indeed, actual mathematical theorems. 

 Seems unlikely to create two spheres of equal dimension to the original sphere. Sounds like a great way to get business out of infinite produce!

 Also that Hairy Ball theory... would love to hear what the mathematical definition of brushie-brushies are!

Oh, no; they're all quite real. The one sphere to two spheres thing involves decomposing the original sphere into an infinite number of sets of points and procedurally recombining them into two duplicate spheres of the same volume. This is an elaborated result of the counterintuitive fact that the number of points in one sphere is the same as the number of points in two spheres - aleph-nough (a particular size of infinity).

The hairy ball theorem, on the the other hand, is a rather abstract result that says that if a surface topologically equivalent to a sphere is covered with a continuous field of horizontal vectors, there must be at least one vortex formation. This can be roughly simplified to the idea that if you have a sphere or similar shape with hair covering its surface, you can't comb it all flat without at least one cowlick, and also ties into the fact that the wind patterns across the Earth have to include at least one spiral formation/hurricane. Maths has bizarre generalisations.