SECOND PART of my answer.
(The ultimate bottom)
You find a contradiction between placing indivisible atoms of spacetime at the bottom of reality, and the need for a digital computer that runs the evolution of this collection of atoms. You seem annoyed by the fact that such a digital computer would represent a 'deeper background structure' beneath the level of these indivisible spacetime atoms, requiring perhaps even an operator (you ask 'who/what manages this digital computer'?): discrete spacetime would no longer be the very 'bottom' of the universe. The answer is simple: I do NOT postulate the existence of such a computer, in the basement or elsewhere, as clearly written at the bottom of page 1 in my essay. An algorithm, as well as a differential equation, is simply a formal way to describe dynamics. No need for hardware.
(Computation and curvature)
You write that 'at the center of a black hole, a digital computation is not possible because spacetime curvature becomes infinite'. I completely agree that your PC (or even my Mac!) would start having computing problems a while after crossing a black hole horizon. But, again, we are not talking about hardware, we are abstractly talking about computation. Better: computations on graphs. The variety of structures and phenomena that one can obtain out of algorithmically evolving graphs is formidable. A cheap proof of this, if you wish, is that when you describe phenomena such as those involving black holes, you tend to visualize things precisely in terms of points and arrows, which is what directed graphs are made of...
By the way, a very good 1999 paper by Margenstern and Morita proves that, in the context of cellular automata, spatial (negative) curvature offers indeed a great computational advantage over flat space (M. Margenstern, K. Morita, 'A Polynomial Solution for 3-SAT in the Space of Cellular Automata in the Hyperbolic Plane', Journal of Universal Computer Science, vol. 5, no. 9, Springer, 1999, pp. 563-573). Amazing.
However, to me, the appropriate question is not whether a computation is possible inside a black hole, but, rather, what IS a black hole, how does it look like or manifest, in a graph-based, computational spacetime. I don't know. But simple concepts such as sink node (one with only incoming arcs), or strongly connected components are available that may help. Curvature can also be defined for (planar) graphs, called 'combinatorial curvature'. It is only finite, but possibly unbounded, if you grow the number of faces sharing the inspected node, as it can indeed happen in some of my algorithmic causal sets.
(Quantum effects)
If I had good answers for these problems, they would have appeared very early in the essay. Stephen Wolfram has some potentially useful suggestions for entanglement (NKS book, ch. 9). I have long discussed the quantum effects issue with Alex Lamb, who is also participating to this Contest , and he half-managed to convince me that a form of nonlocality could be achieved if we imagine the algorithmic graph rewriting to take place directly on the causal set, rather than on an underlying spatial support, as I've done so far.
But the more general question is: are we going to eventually apply the standard QM techniques and compose instances of discrete spacetime in a gravitational path integral -- a sum over histories? Perhaps... but later. In doing so, we could follow, for example, the work of Renate Loll and collaborators, who take sums of causal dynamic triangulations (CDT) of spacetimes, and investigate consequences such as emergent spacetime dimension. But I am reluctant to do this, at least before having fully explored the potential of a classical approach to emergence in discrete, computational spacetime. Exciting phenomena such as those illustrated at Figures 3 - 5 of my essay would probably be obscured by a QM treatment.
Finally, while I have no problems in conceiving 'computations' without 'computers', I am more skeptical about defining 'observations' without 'observers', and QM effects do depend on rather intrusive observers, engaging in interactions that affect both the observed and the observer subsystem. In a tiny, discrete, newborn universe that has just reached the size of say 33 elements -- counting edges, nodes, faces, or simplices -- is there enough room for this interaction? Is there room for an observer? (an INTERNAL observer, that is). Maybe quantum effects, and perhaps even relativistic ones, unfold only at a later stage, when entities emerge that can play the role of (proto-)observers. But this is only wild speculation.