One of the results due to Georges Glaeser is the following: there exists a non-negative $C^\infty$ function $f$ on the real line, flat at its zeroes, such that $\sqrt{f}$ is not $C^2$. On the other hand, $\sqrt{f}$ is $C^1$ for any such $f$.

Question 1: is it possible to find $f$ as above such that $\sqrt{f}$ is not twice differentiable at a point? In Glaeser's counterexample mentioned above, $\sqrt{f}$ is twice differentiable with an unbounded second derivative.

Question 2: is it possible to find $f$ as above such that there is no function $g$, $C^2$ on the real line such that $f=g^2$. Here $g$ is allowed to take negative values, which is not the case of $\sqrt f$.

Question 3: is it possible to find $f$ as above such that there is no $C^\infty$ function $g:\mathbb R\longrightarrow \mathbb C$ such that $f=\vert g\vert^2$.