They are a bit misrepresented. Think of most of them as "we know today's theory of physics breaks beyond these parameter regimes". There is no "minimal distance", rather there is "distance at which our math model does not work". No need to assume that things go discrete at that distance. There are some that are in a different style though: v<c is a fundamental postulate (stated in a very cluncky way).

I think not for distance and time. I’m not a physicist but Nima Arkani-Hamed phrases the minimal distance measurable as a physical process limited by QM and GR effects. To probe smaller and smaller distances one needs more and more energy, and past some minimum distance the energy becomes so great your experiment collapses into a black hole. That’s just what QM (small wavelength->high energy) and GR (force of gravity compared to the other forces, enough energy and gravity dominates) tell us. Since energy is related to time by a symmetry I think minimal time is implicated in the same way. A better interpretation than minimal discrete values of the universe is we are butting up against experimental limitations. An old concept that doesn’t necessarily mean the world is limited in the same we way are.