How many times is the composition of multivariable functions partially differentiable?

Question

Let $m, n, p \in \{1,2,\dots\}$, let $q, r \in \{0,1,\dots\}$, let $D$ be a non-empty, open subset of $\mathbb{R}^m$, let $E$ be an open subset of $\mathbb{R}^n$, let $f : D\rightarrow E$ be $q$ times partially differentiable, and let $g : E\rightarrow\mathbb{R}^p$ be $r$ times partially differentiable.

1. What is the greatest lower bound $s \in \{0,1,\dots\}\cup\{\infty\}$ on the number of times $g\circ f$ is necessarily partially differentiable?

2. Suppose, moreover, that $f$'s $q$-degree partial derivatives and $g$'s $r$-degree partial derivatives are all continuous. Are $g\circ f$'s $s$-degree partial derivatives continuous (assuming $s<\infty$)?

Answer 1

It's a partial answer, but I've found the following proposition on p. 58 of James Munkres' *Analysis on Manifolds", Westview Press (1991):

Let $A$ be open in $\mathbb{R}^m$; let $B$ be open in $\mathbb{R}^n$. Let $f:A\rightarrow\mathbb{R}^n$ and $g:B\rightarrow\mathbb{R}^p$, with $f(A) \subseteq B$. If $f$ and $g$ are of class $C^r$, so is the composite function $g\circ f$.

The symbol $C^r$ is defined on p. 52 as follows.

Suppose $f$ is a function mapping an open set $A$ of $\mathbb{R}^m$ into $\mathbb{R}^n$, and suppose the partial derivatives $D_jf_i$ of the component functions of $f$ exist on $A$. These then are functions from $A$ to $\mathbb{R}$, and we may consider their partial derivatives, which have the form $D_k(D_jf_i)$ and are called the second-order partial derivatives of $f$. Similarly, one defines the third-order partial derivatives of the functions $f_i$, or more generally the partial derivatives of order $r$ for arbitrary $r$.

If the partial derivatives of the functions $f_i$ of order less than or equal to $r$ are continuous on $A$, we say $f$ is of class $C^r$ on $A$. [...] We say $f$ is of class $C^\infty$ on $A$ if the partials of the functions $f_i$ of all orders are continuous on $A$.