Let $A\subset \mathbb{R}^n$ be a bounded open subset. The cut locus $C$ of $A$ is basically defined as the domain of nondifferentiability of $dist_{\partial A}: A\to \mathbb{R}$. Blum's Medial Axis Transform (MAT) is the set $$M(A):=\{x\in A ~|~  card|argmin_{y\in \partial A} dist(x,y)|\geq 2\}.$$ We find $M(A) \subset C$, although $M(A)$ is not always closed. We find $\overline{M(A)}=C$.


(1) We find both the cut locus $C$ and $M(A)$ have the homotopy type of the domain $A$, so if $A$ (or $\Omega$ in the notation of the OP) is contractible, then $C$ and $M(A)$ is contractible. This is key property of Blum's Medial Axis Transform. Constructing a strong deformation retract of the domain onto $C$ is reasonably easy: map every point $x$ to the centre $m(x)$ of the maximal round disk containing $x$ and contained in $A$. (Of course, you need prove this maximal disk is unique and varies continuously with $x\in A$).


(2) The OP wants an explanation for why $C$ is a "locally finite graph". Now strictly speaking neither $C$ nor $M(A)$ are *graphs* when $A$ is two-dimensional. Rather they are subsets with $1$-dimensional and $0$-dimensional components. E.g. they do not have the form $(V,E)$ for a set of "edges" and "vertices", etc. They are a type of singular subset representing all the *connectivity* information of $A$. 

Nonetheless, our response to why $C$ is locally finite is this: *if* there was a point $x\in C$ or $x\in M(A)$ which had infinite *degree* (i.e. there were infinitely many branches intersecting at $x$), then the maximal disk centred at $x$ would intersect the boundary $\partial A$ at infinitely many points. But for what sets $A$ do disks have infinite intersection? Those sets $A$ whose boundaries $\partial A$ have circular arcs, e.g. disks, or unions of disks, or polygons with special rounded corners. However this leads to contradictions, because the cut locus of disks and arcs, etc., consists of points. And the "degree" of the "graph" at that point becomes zero. This is example of the upper semicontinuity of $M(A)$ and $C$: you can construct sets $A_k$ where the degree of the vertex is becoming arbitrarily large, however in the limit the degree collapses to zero. For example, consider $A_k$ the $k$-sided polygon which limits to the round disk $A_\infty$.

(3) The MAT (or cut locus) is typically trivalent (all vertices degree 3).  But arbitrarily small deformations can significantly increase the number of edges in $C$, c.f. instability and discontinuity properties of MAT (http://midag.cs.unc.edu/pubs/papers/IJCV03-Katz-BlumMedAxis.pdf).

If the boundary is such that every intersection with a disk is finite, then $C$ will be locally finite. Again, only round boundaries which contain circular arcs will have the property that disks intersect the boundary in infinitely many points, but these do *not* lead to infinite degree in the graph.