Q1: Let $M, N$ be smooth manifolds and $f: U \rightarrow V$ be a continuous map. There are charts $(U,h)$ of $M$ and $(V,k)$ of $N$ and $p \in U$ such that $k \circ f \circ h^{-1}$ is differentiable at $p$. Is $f$ differentiable at $p$.

A1: Yes, by definition it is.

A2: No, there could be charts where $k \circ f \circ h^{-1}$ is not differentiable at $p$.

A3: One needs more information.

Q2: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a $C^\infty$-function. Is it also a $C^\infty$-smooth maps between manifolds?

A1: Yes, by using the standard smooth structure of $\mathbb{R}^2$.

A2: No, it can never be differentiable as a map between manifolds.

A3: One needs more information.