Computability of function composition
Suppose that $f(x) = 0$. Then $g(f(x))$ is computable for any function $g$, computable or not. This explains 1.
As for 2, the composition of two computable functions is computable, so if both $f$ and $h$ are computable, so is $g$. Since $f$ is computable and $g$ isn't, the only conclusion is that $h$ is not computable.