Differentiability and integrability of a function composed with itself

Question

I am reviewing for an exam and came across a multi-part question that I am having a hard time with. We are asked to prove or disprove the following statements. If a statement is false, what additional hypothesis would make it true? (Note: $f^{\circ 2}=f\circ f$).

Let $f:[0,1]\rightarrow [0,1]$ be a continuous function.

1. If $f$ is differentiable, then so is $f^{\circ 2}$.
2. If $f^{\circ 2}$ is differentiable, then so is $f$.

Let $f:[0,1]\rightarrow [0,1]$ be a function (not necessarily continuous).

3. If $f$ is Riemann integrable, then so is $f^{\circ 2}$.
4. If $f^{\circ 2}$ is Riemann integrable, then so is $f$.

I think statement 1 is as simple as saying that if $f$ is differentiable, then $f^{\circ 2}$ is the composition of differentiable functions and is thus differentiable. Any guidance would be much appreciated!