Since youre in euclidean space, i will speak of medial axis transform rather than cut locus.
(1) The cut locus has the homotopy type of the domain, so if $\Omega$ is contractible, then $C$ 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 $x$.
(2) The MAT (or cut locus) will be locally finite graph if the boundary $\gamma$ is smooth (say $C^2$). But arbitrarily small deformations can significantly increase the number of edges in $C$, c.f. instability and discontinuity properties of MAT ("Untangling Blums MAT").
(3) $C$ has as many "branches" as disks have tangency points with the boundary $\gamma$. So if the boundary is such that every intersection with a disk is finite, then $C$ will be locally finite. This is fairly "obvious" though a strict formal proof might be harder to pinpoint.