  We investigate the interplay of the following regularity properties for
non-simple C*-algebras: finite nuclear dimension, Z-stability, and algebraic
regularity in the Cuntz semigroup. We show that finite nuclear dimension
implies algebraic regularity in the Cuntz semigroup, provided that known type I
obstructions are avoided. We demonstrate how finite nuclear dimension can be
used to study the structure of the central sequence algebra, by factorizing the
identity map on the central sequence algebra, in a manner resembling the
factorization arising in the definition of nuclear dimension.
  Results about the central sequence algebra are used to attack the conjecture
that finite nuclear dimension implies Z-stability, for sufficiently non-type I,
separable C*-algebras. We prove this conjecture in the following cases: (i) the
C*-algebra has no purely infinite subquotients and its primitive ideal space
has a basis of compact open sets, (ii) the C*-algebra has no purely infinite
quotients and its primitive ideal space is Hausdorff. In particular, this
covers C*-algebras with finite decomposition rank and real rank zero. Our
results hold more generally for C*-algebras with locally finite nuclear
dimension which are (M,N)-pure (a regularity condition of the Cuntz semigroup).
