Recently I have read the paper \emph{Whitney's problem on extendability of functions and an intrinsic metric} written by Nahum Zobin and published by Advances in Mathematics in 1998. I am confused about one proposition of this paper. Now let me state the background of this paper. 

Let $\Omega$ be a bounded connected open set in $\mathbb{R}^n$. Consider the following Sobolev function space
\begin{equation*}
  W^1_\infty(\Omega)=\{f\in C(\Omega):\forall \alpha\in \mathbb{Z}^n_+,|\alpha|=1,f^{(\alpha)}\in L^\infty(\Omega)\}.
\end{equation*}
(This definition follows from E. M. Stein, \emph{Singular Integrals and Differentiability Properties of Functions}, Princeton University Press, 1970, Chap. V, Section 6.2.)
Here $|\alpha|=\sum_{i=1}^n \alpha_i$ for $\alpha=(\alpha_1,\cdots,\alpha_n)\in \mathbb{Z}^n_+$, $f^{(\alpha)}$ denotes the corresponding distributional partial derivative, $C(\Omega)$ denotes the space of continuous functions, $L^\infty(\Omega)$ denotes the space of essentially bounded functions on $\Omega$. 

Let $W^1_\infty(\mathbb{R}^n)|_\Omega$ denote the space of restrictions to $\Omega$ of functions from $W^1_\infty(\mathbb{R}^n)$. 

For $x,y\in\Omega$, let
\begin{equation*}
  d_\Omega(x,y)=\text{infimum of lengths of polygonal paths in $\Omega$ joining $x$ and $y$}.
\end{equation*}
$d_\Omega(x,y)$ is called the \emph{intrinsic metric} in $\Omega$. 

Now fix $x\in \Omega$, consider the function $f(y)=d_\Omega(x,y)$. Then $f\in W^1_\infty(\Omega)$ and 
\begin{equation}\label{*}
  \sup_{\substack{z\in \Omega\\ |\alpha|=1}} |f^{(\alpha)}(z)|=1.
\end{equation}
If $W^1_\infty(\Omega)\subset W^1_\infty(\mathbb{R}^n)|_\Omega$, then $f$ is extendible to $\tilde{f}\in W^1_\infty(\mathbb{R}^n)$ and the open mapping theorem guarantees that on can choose $\tilde{f}$ such that 
\begin{equation}\label{**}
  \sup_{\substack{z\in \mathbb{R}^n\\ |\alpha|=1}} |f^{(\alpha)}(z)|
  \leqslant C\sup_{\substack{z\in \Omega\\ |\alpha|=1}} |f^{(\alpha)}(z)|=C. 
\end{equation}
So,
\begin{align}
  d_\Omega(x,y)&=|f(y)-f(x)|=|\tilde{f}(y)-\tilde{f}(x)|\notag\\
  &\leqslant \sup_{\substack{z\in \mathbb{R}^n\\ |\alpha|=1}}|\tilde{f}^{(\alpha)}(z)|\cdot |x-y|
  \leqslant C|x-y|.\label{***}
\end{align}

My questions are:

(1) Why $f\in W^1_\infty(\Omega)$ and \eqref{*} holds?

(2) How to use the open mapping theorem and why \eqref{**} holds?

(3) Why the first inequality of \eqref{***} holds? 

About Question (1), I know $f$ is Lipschitz with constant $1$ under the intrinsic metric, that is, 
\begin{equation*}
  |f(z_1)-f(z_2)|\leqslant d_\Omega(z_1,z_2),\quad \forall z_1,z_2\in\Omega.
\end{equation*}
But I don't know whether this property can help us deal with this question. 

About question (3), I know now $\tilde{f}$ is Lipschitz continuous on $\mathbb{R}^n$ under the usual Euclidean metric, but I don't know how $\sup_{\substack{z\in \mathbb{R}^n\\ |\alpha|=1}}|\tilde{f}^{(\alpha)}(z)|$ appears.