When can a freely moving sphere escape from a 'cage' defined by a set of impassible coordinates?

Replace the pins by balls of radius $R_{ball}$ and the ball by a point. This is a logically equivalent formulation. The question, then, is: given a finite set of balls, $B_1$, $B_2$, ...., $B_k$ in $\mathbb{R}^n$, and a point $x$, how to determine where $x$ is in the unbounded component of $\mathbb{R}^n \setminus \bigcup B_i$.

I don't know the answer to this, but here is an easy way to compute the number of connected components of $\mathbb{R}^n \setminus \bigcup B_i$. In other words, I can determine whether there is some place from which a ball cannot escape.

By Alexander duality, the number of bounded components of $\mathbb{R}^n \setminus \bigcup B_i$ this is the dimension of $H_n(\bigcup B_i)$.

Cover $\bigcup B_i$ by the $B_i$. Every intersection of finitely many $B_i$ is convex, hence contractible. So $\bigcup B_i$ is homotopic to the nerve of this cover. That is a simplicial complex, so it is easy to compute its homology.

One final practical idea: I have used painting software where I could click on a point and it would color every point which was connected to that one. Maybe the algorithms used to make that software could solve this problem as well?

I recently heard a beautiful talk by Yuliy Baryshnikov on the general question of when an object can be pinned by some set of fixed points. They consider arbitrary objects in 2D and prove the following theorem:

Let D be a planar domain. Either one can pull a configuration C of two points $\{p_1,p_2\}$ around D, or there exists a full rotation of C entirely within D, that is a loop π′: $S^1$ → E (E being the Euclidean group of transformations) such that the vector $π′\circ p_1 − π′ \circ p_2$ turns around the origin (perhaps, several times).

They use a topological approach which uses Mayer-Vietoris sequences in homology; apparently to generalize to 3D one must use Mayer-Vietoris spectral sequences, though this is "future work".

The slides are here and do include some discussion of computing the possibility of caging / linking effectively, but again, they focus on the 2D problem.