Suppose I have finite list of $n$ balls, specifying their positions and radii. The balls can have non-empty intersections.
Is there an algorithm to compute the volume of the region resulting from the union of all $n$ balls? How hard is this problem?
Is there a source code out there that does the computation already?
Yes. there is an algorithm with time complexity $O(n)$. Details of implementation and algorithm step in this article: Computing the Volume of a Union of Balls: A Certified Algorithm which is implemented in CGAL. For example see this or this as a reference.