After scrutinizing the literature some more, I've concluded that, in the case when the causal structure as defined in Willie's answer comes from a Lorentzian metric, the answer to my question has been available for some time and more recently in the general case.

The main point of confusion, and the reason I did not realize this sooner, is that that the property of a domain being *lens-shaped* is usually expressed in different terms. Namely, for a Lorentzian manifold $M$, the property of being lens-shaped is equivalent to the existence of a smooth time function $f$ (it increases along every future directed timelike curve and the image of the composition of each such curve with $f$ is the whole real line $({-\infty},\infty)$) such that each level set is a Cauchy surface diffeomorphic to $S$. In other words, the spacetime smoothly and causally factors as $M\cong \mathbb{R}\times S$. Adding a boundary to compactify $S$, if necessary, and rescaling $f$ to make sure its range is $({-1},1)$, it is easy to see that this factorization (or splitting) is equivalent to the property of being lens-shaped, as defined precisely in Willie's answer.

So, the answer part (a) of my question is subsumed by the well known equivalence of global hyperbolicity of $M$ (as defined in terms of timelike curves) to the existence of the smooth causal splitting $M\cong \mathbb{R}\times S$, with $S$ diffeomorphic to a Cauchy surface of $M$. Part (b) is then answered by noting that both $D(S)$ and the union of all lens-shaped domains are maximal under the respective conditions of global hyperbolicity and being lens-shaped and again using the fact that these conditions are equivalent.

Now, the smooth causal splitting property was first established by Geroch ([JMP][1], 1970), though only for Lorentzian manifolds and the splitting was only show to be topological. Again for Lorentzian manifolds, the smoothness was established more recently by Bernal and Sánchez ([CMP][2], 2003). Finally, for more general causal structures as defined by Willie's answer, smooth causal splitting was established very recently by Fathi and Siconolfi ([Math Proc CPS][3], 2011), incidentally using quite different methods from the previous work.

The information above definitely answers my original question. However, if I could, I would mark this answer as correct jointly with the one given earlier by Willie, as it certainly helped greatly.

  [1]: http://dx.doi.org/10.1063/1.1665157
  [2]: http://dx.doi.org/10.1007/s00220-003-0982-6
  [3]: http://dx.doi.org/10.1017/S0305004111000661