This is true and it follows from the following result:

Now you can exhaust $X$ by compact convex subdomains $W_k\Subset X$, $X=\bigcup_k W_k$. Then $f$ is convex and Lipschitz on $W_k$. Extend $f$ from $W_k$ to a convex function $\tilde{f}_k:\mathbb{R}^n\to\mathbb{R}$, $\tilde{f}_k=f$ on $W_k$, and apply the result you mention to each of the functions $\tilde{f}_k$.

The above lemma is well known. It is copied here from:

D. Azagra, P. Hajłasz, Lusin-type properties of convex functions and convex bodies. J. Geom. Anal. 31 (2021), 11685–11701.

The theorem about the size of the set of non-differentiablity points of a convex function is actually due to Zajíček who proved a much stronger result is 1979. For comments and references, see https://mathoverflow.net/a/354985/121665.

This is true and it follows from the following result:

Lemma. Suppset that $f:W\to\mathbb{R}$ is convex and $L$-Lipschitz, where $W\subset\mathbb{R}^n$ is convex. Then, $$ \tilde{f}(x)=\inf_{z\in W}\, \big(f(z)+L|x-z|\big), \quad x\in\mathbb{R}^n $$ is convex and $L$-Lipschitz on $\mathbb{R}^n$, and $\tilde{f}=f$ on $W$.

Now you can exhaust $X$ by compact convex subdomains $W_k\Subset X$, $X=\bigcup_k W_k$. Then $f$ is convex and Lipschitz on $W_k$. Extend $f$ from $W_k$ to a convex function $\tilde{f}_k:\mathbb{R}^n\to\mathbb{R}$, $\tilde{f}_k=f$ on $W_k$, and apply the result you mention to each of the functions $\tilde{f}_k$.

The above lemma is well known and you can find it for example in:

D. Azagra, P. Hajłasz, Lusin-type properties of convex functions and convex bodies. J. Geom. Anal. 31 (2021), 11685–11701.

The theorem about the size of the set of non-differentiablity points of a convex function is actually due to Zajíček who proved a much stronger result is 1979. For comments and references, see https://mathoverflow.net/a/354985/121665.