Dear Tomasso,
I will try to answer your questions.
First of all, let me note that in my understanding, the question "Is Reality Digital Or Analog?" is meaningful only if it is understood as a question about mathematics describing reality. Some contest participants argue that e.g. mathematics might be continuous but physics - discrete but I don't understand such arguments. In my essay I argue that if we accept a principle that only those statements are meaningful, which can be experimentally verified (at least in principle) then only a finite mathematics can describe reality. We have no experience in this field and so nothing can be stated for sure.
But if indeed only a finite math describes reality then it is reasonable to think that our Universe is finite. Indeed, in finite systems, consistent calculations can be performed only modulo some number. So if we find effects which can be explained only by finite math then it will be a strong argument that the Universe is finite. You mention a possibility that the Universe is infinite but a finite math gives a good description of reality in some areas.
Probably this possibility is not realistic since if the Universe is infinite then it is not clear why physics is described by a finite math, but of course our experience is not sufficient and maybe for some reasons this scenario takes place.
If we try to construct a quantum theory based on a finite math then probably many possibilities can be investigated. For example, I argue that standard division has a limited meaning but I also do not see why division in Galois fields has a fundamental meaning. As I note in my essay, Metod Saniga believes that a theory based on a finite ring is even more interesting. But technically it is convenient to work with a field; for example, a well known result in algebra is that the dimension of a linear space is well defined only if the space is over a field or body. I show that a case of the field with p elements contradicts experiment and so the field should be extended. A simplest extension is a Galois field F(p^2) with p^2 elements but of course we cannot exclude a possibility that there are reasons for a Galois field version of the theory where the field is more complicated that F(p^2) and the latter is only an approximation.
It is easy to show (see e.g. pp. 5 and 6 of my essay) that there exists a correspondence principle between projective complex Hilbert spaces and projective spaces over Galois fields F(p^2). However, even in this case GFQT and standard theory are considerably different. For example, in GFQT one irreducible representation of the symmetry algebra describes a particle and its antiparticle simultaneously and there are no neutral elementary particles (so even the photon cannot be elementary). These problems are discussed in [11], which can be found at http://www.mdpi.com/2073-8994/2/4/1810/ . I have also considered a hypothesis that gravity is simply a manifestation of finiteness of physics and this work is underway (see e.g. http://xxx.lanl.gov/abs/0905.0767 ). In summary, the key problem is to construct a theory relating GFQT with experiment and this cannot be done simply by analogy with standard theory. If this problem is solved then it will be clear whether nature is described by finite math or not.
You also ask whether p is a fundamental constant or it may change with time. In a wider context, I am aware of different opinions on GFQT. Mathematicians sometimes say that a version with only one Galois field is not attractive since it is not clear why a special value of p is chosen. For this reason some of them prefer an adelic approach but in this case we do not have finiteness. On the contrary, physicists typically believe that no new fundamental constant p is needed. I believe that this is rather strange since history of physics tells us that new theories arise when a parameter, which in the old theory was zero or \infty becomes finite. Indeed, classical physics has no parameters at all but relativity arises when c is not infinite but finite. Analogously, quantum theory arises when \hbar is not zero but finite. So I believe that it is rather attractive that GFQT arises when p is not infinite but finite. My assumption is that p is the characteristic of our Universe since it defines the laws of physics. If this is reasonable then p is a fundamental constant and it is reasonable to believe that it does not change with time. I believe that such dimensionful quantities as the gravitational and cosmological constants are not fundamental and it's reasonable to think that they are changing with time. The arguments and my opinion on c and \hbar are given in [9], which can be found at http://www.mdpi.com/2073-8994/2/4/1945/ .
Finally, as far as your question about 10^234 atoms of spacetime is concerned, I would like to note that I fully agree with Heisenberg and others that a fundamental physical theory should not involve spacetime at all. A detailed description of my point of view can be found in [9,11].
Best regards, Felix.