Let $X\subset \mathbb{R}^N$ be a convex cone (i.e., for all $x,y\in X$ and $\alpha,\beta\geq 0$ scalars, $\alpha x+\beta y\in X$). Define the set $$A(x)=\{a:x+a\in X \wedge x-a\in X\}.$$
Then, $A(x)=(-x+X)\cap (x+(-X))$, where this set addition has the standard definition. Is this claim true?
Yes, it's true. It's simply due to the basic properties of set addition. That is, since $x + a \in X \Leftrightarrow a \in -x + X$ and $x-a \in X \Leftrightarrow a\in x + (-X)$, we have:
$$ \begin{aligned} A(x) &= \{a:x+a\in X \wedge x-a\in X\} \\ &= \{a:a \in -x + X \wedge a\in x + (-X)\} \\ &= (-x+X)\cap (x+(-X)) \end{aligned} $$ The fact that $X$ is a convex cone is irrelevant here.