Monotonic?

f must be bijective

and every bijective continuous function from [0,1] to [0,1] is monotonic

ya. but we only need it to be injective and continuous, and for that note that if f(x₁) = f(x₂) then f(f(x₁)) = f(f(x₂)) which implies x₁=x₂

Which theorem allows us to relate this fact to f being monotonic?

we have f'(f(x))f'(x)=1

and f'(f(x))=\lim_{h\rightarrow 0}\frac{f(f(x+h))-f(f(x))}{h}

and f(f(x))=x for all values x in the given interval so f'(f(x))=1

so from the first eqn we have f'(x)=1

careful, nothing has been said about differentiability.

The last result in #3 easily follows from intermediate value property.