Is the universe finite, both in the sense of being a closed spacetime manifold, as viewed from the macro level, but also in the sense of being fully discrete and finite in all of its intricate quantum level construction?
The most popular current research, i.e. string theory, builds a machinery on top of a continous notion of spacetime. Would a more accurate model of the physical world have to build on top of a finite, discrete, network like, relational spacetime model?
Your question is good, but dangerously edgy even to try to answer. Alas, since I seem at times I'm prone to trying rather than doing nothing at all... :)
Let me suggest an intentionally different way of approaching your question: Only conservation is absolute. Both continuous and discrete behaviors are approximate and mutable expressions of the absolute conservation of certain quantities.
I'll point to the curious mix in quantum theory of continuous wave functions and discrete outcomes as a possible example. The most accurate way to represent a wave function mathematically is as precisely continuous, yet that same continuous perfection can only be accessed experimentally in terms of discrete result that sample many such nominally perfect wave functions. But the fully discrete particle view never fully wins either, since for example detecting an absolutely positioned particle is a physical impossibility in our universe. There is instead a sort of "bounce point" between the two views, one whose scale is captured by Planck's constant.
But what does always apply without exception in analyzing quantum problems, even across light years of separation in cases of entanglement,[1] is the absolute and unyielding conservation of a certain small set of properties that includes mass-energy, charge, momentum, spin, and a few more obscure quantities such as $T_3$. So why not just declare these conservation rules to be the real absolutes, with the variable interplay we observe between continuous and discrete views as more of an emergent perspective on how the conservation rules play out over time?
So: Since you asked a good but highly speculative question, I hope readers of this answer will have some mercy on me for giving an answer. While I don't think my answer is exactly radical -- few would debate the importance of the absolute conservation laws in physics, I think! -- I fully admit that it is highly speculative in terms of the priorities I am suggesting.
[1] Focusing on conservation first puts entanglement in a rather different light. It suggests that far from being an odd or minor side effect of QM, entanglement at the classical level reflects the unresolved remnants of deeper conservation laws that mostly work themselves out into something we call "locality of effect" when they are expanded out in a self-consistent fashion over that curious dimension we call time. By "time" in this context I mean the classical, entropic, macroscopic time we know on a daily basis. The quantum version of time, the wonderfully symmetric one, occurs when one or more of those absolute conservation laws insists on keeping its options open. That openness, expressed as the uncertainty principle, makes the irreversible time we know a lot less relevant at the quantum level.
My concept of motion is that one moves from one point to its successor which implies the countability and discontinuity of points. Like Zeno, I find the idea of applying Real numbers which are uncountable to conflict with this concept of motion. The question does arise as to what is present during the intervals in discrete time systems. As I have come to accept that there is superposition of "worlds" all but one of which are unobservable by agents such as us, I have also come to accept that agency itself is unobservable, being only the observer and is present during the gaps in time. A side note such a system of superpositioned "worlds" is entangled with agents for the life time of the agents. Whether there is but one agent who sequentially follows the multiple entanglement paths through superpositioned worlds or separate agents for each such path seems simpler if it is the former.
