Let $f: \mathbb R^n \rightarrow \mathbb R$ and $g: \mathbb R^n \rightarrow \mathbb R$ be continuously differentiable functions whose sublevel sets are compact. Suppose that $\nabla f(x) = 0$ for some $x \in \mathbb R^n$ if and only if $x$ is a global minimum of $f$. Suppose the same for $g$. Then, is it true that $\nabla(f + g)(x) = 0$ for some $x$ if and only if $x$ is a global minimum of $f + g$?
It seems to be true if both $f$ and $g$ are convex. What if one of them is not convex?
This is not true.
Your characterization of $f$ and $g$ require that they are so-called invex functions, see Theorem 1 of this reference.
However, invex functions are not closed under addition. As an example, take
$$ f(x)= \begin{cases} 0.02 \log(-10x) + 0.01, \:\: \text{if } x < -0.1\\ x^2, \:\: \text{if } -0.1 \leq x \leq 0.1\\ 0.02 \log(10x) + 0.01, \:\: \text{if } x > -0.1 \end{cases} $$
and $g(x) = 0.01(x-3)^2$. The function $f$ is invex and the function $g$ is convex and they satisfy your other requirements, but $f+g$ is not invex (see image below; $f$ and $g$ are shown using solid lines and $f+g$ is shown using dotted lines).
An example with $g$ also nonconvex is
$$ g(x)= \begin{cases} 0.02 \log(-10(2-x)) + 0.01, \:\: \text{if } x < 1.9\\ (x-2)^2, \:\: \text{if } 1.9 \leq x \leq 2.1\\ 0.02 \log(10(x-2)) + 0.01, \:\: \text{if } x > 2.1. \end{cases} $$
Both $f$ and $g$ are invex and satisfy your requirements, but $f + g$ is not invex.