..experiments could have more than one interpretation or even conclusion..
Well, exactly. If some experiment can lead to the more theories, it means, all these theories are valid as well from their perspectives. For example, I can describe the gravitational lensing both from relativity perspective (where the space deforms and the light speed remains constant), both from Aether theory perspective (where the space metric is fixed and the light speed doesn't remain constant). And both these models are correct, in fact - they just describing the reality from different perspective (inner/outer in particular). Therefore to discuss differences between theories has no meaning - we should target to their similarities, to their common message which can lead us to the more general level of understanding. It requires to cooperate, not to fight.
..How can you be sure that space and time at the level of the elementary particles are continuous?..
We cannot, but the space continuity concept is necessary with respect to causality. We should consider, every piece of space between particles is filled by density gradients of another particles, recursively to avoid the "spooky action" at distance. At the case, the difference between densities/sizes of particles is large, it can lead to the illusion of empty space. But for example the wave character of energy spreading is demonstrating, here must be some "elastic" environment. Without it the elastic properties of common matter could be explained by elastic properties of space, which isn't apparently correct. The elasticity and wave spreading are fundamental attributes of common matter, because their constants (permitivity, permeability...) are depending on properties of particles, not the space, which fills the gaps between particles. The space as such has no attributes, which could be changed from experiment to experiment.
