Dear Luca,
Here are some answers before I read you prose.
1. Indeed, I did not mention quantum mechanics because this elaborated theory is `at the leaves of the tree', while I really start from the base of the trunk (and even try to show some roots).
However, with more room available, an interesting remark was possible. I noted that mathematically, any structure is characterised by its invariants. An accomplished theory is axiomatised; and the structure of the axioms carries its invariants in all the further developments. Gilles Cohen-Tannoudji gives an illuminating interpretation of universal constants of physics in terms of such invariants, for which he uses concept of horizon, inspired by Ferdinand Gonseth (see his contribution, alongside those of Bohr, Einstein, de Broglie, Heisenberg, Reichenbach notably, in Dialectica 2:3/4 (1948) pp. 305--424).
Now, the very first hypothesis of physics expels the subject (cognitive subject). That applies to all physics, with consequent limitations I have pointed at. It could only happen that in some cases, it would become absolutely necessary to reintegrate the observer. That has happened in quantum mechanics. The most spectacular illustration is the von Neummann-Wigner interpretation of quantum mechanics or any of its variants, whereby to make it short, consciousness causes the collapse. (This is not the place to discuss whether this interpretation is right or not. Suffices to say that many prominent actors of physics took it seriously, and that its core still holds.) My comment is that it is no surprise that when it becomes absolutely necessary to take the subject into account --and this is very closely connected to measurement-- he suddenly lands in the middle of the discourse. (The close connection to measurement is clearly seen in quantum mechanics, and this is why in the essay I have mentioned the relative nature of any measurement, and the formal closeness between measurement, modelling, perception, and formalisation, which is connected to the subject making categories --symmetries--, and going from what is given in the theory as a description at infinite resolution --in real numbers--, to a reduced, categorised view --the transition to macroscopic in a quantum mechanics measurement.)
So there could be a criticism addressed to `quantum mechanics' --say von Neumann: that he should have explicitly stated that a subject was to be added as an additional basic element, to his axioms of quantum mechanics, from the moment he was summoned to effectively make something happen, to collapse the wave function.
Adding the subject is basically what I do. But in the same movement, I find unnecessary to carry with me all the usual established axioms, I proceed straightforwardly by restarting from scratch.
2. With these topics of perception, measurement, I have a natural transition to time, your next question, with this quotation of Heisenberg, in an interview by Paul Buckley, in Glimpsing reality: ideas in physics and the link to biology:
PB--How does quantum mechanics deal with time flow or does it in fact say anything at all about it?
WH--I would have to repeat what C. von Weizsäcker said in his papers: that time is the precondition of quantum mechanics, because we want to go from one experiment to another, that is from one time to another. But this is too complicated to go into in detail. I would simply say that the concept of time is really a precondition of quantum theory.
It makes strikingly clear that Heisenberg would never have considered non-existence of time as a conceivable consequence of any theory of physics, for the very reasons I have raised in the essay, and that his position is just one step away from subjective relativity.
3. As to whether time is discrete or continuous (is perceived as, or represented as), I would refer to the definition of a perceptive continuum by Poincaré (he writes physical continuum), which is mentioned in a previous answer of mine:
It has, for instance, been observed that a weight A of 10 grammes and a weight B of 11 grammes produced identical sensations, that the weight B could no longer be distinguished from a weight C of 12 grammes, but that the weight A was readily distinguished from the weight C. Thus the rough results of the experiments may be expressed by the following relations: A = B, B = C, A < C, which may be regarded as the formula of the physical continuum.
The reported effect is quite common, and works as well for mere categories (not only something that purports to a scalar, as in the given example). It works, e.g., for our concept of species: a man is not a monkey, however, if men beget men, and monkeys beget monkeys, we admit since Darwin that man is without interrupt in the descent of monkeys. So monkey = monkey = [...] = man, man ≠ monkey. Of course that property conflicts with objects we can calculate with. (There is a contradiction.)
To sum up, the world is perceptively a continuum. (Perceived) space is continuous, (perceived) time is continuous. But as explained in the essay, a continuous, homogeneous space is only apprehended by its discontinuities: you could not measure a span of space without objects in it, breaking is homogeneity, breaking its invariance under displacement. This is completely consistent with the fact that you cannot see, or feel any sort of space, except by the object that are obstacles to your sight, or your movement, any sort of sensitivity. As for time, these singularities are called events. (Thom has made interesting discussions on saillances and prégnances, discrete objects out of a continuum, but not up to that stage.)
However, as soon as I want to speak about anything, time for instance, I have to pass through categorisation. If, to make it really clear, I put it in writing, I shall attach symbols to categories, and the discrete nature of my discourse will be clearly reflected in a succession of symbols on a paper.
Hence time is discrete as long as it is put in most elementary, formal representation: writing. But here is an example where you can see time, at once both continuous and discrete: when you see time [recorded] in a cliff of sedimentary rocks, you have a basic continuum, within which you can delimit discrete layers (that your geologic eye will interpret as sedimentary events).