Q2) has a negative answer. Namely, if, e.g., $g(x)=-x$ for all $x\in\mathbb{R}$, then there is no continuous $f:\mathbb{R\rightarrow\mathbb{R}}$ such that $f\circ f=g$.
Q2) has a negative answer. Namely, if, e.g., $g(x)=-x$ for all $x\in\mathbb{R}$, then there is no continuous $f:\mathbb{R\rightarrow\mathbb{R}}$ such that $f\circ f=g$.