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According to the calculus,if this size is at the scale of Planck than LHC is sufficient if it is much smaller it is not




The plank scale is 10^19 GEV; LHC will produce at most 14 TEV or roughly 10^4 GEV.

Hence LHC (nor any future planned accelerator) will have enough energy to probe these lengths.

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micro black holes "




There is a very, very, very, very small chance of this happening and no scientist at CERN seriously expects it to happen. And even if they did, this would be a confirmation of relativity and not String Theory (ie: micro black holes are fully explained with current theory).

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A portion of energy could , apparently, disppear in the hidden dimensions





It "could". But then again this has been a possibility since the first collider and in fact is a possiblity in every experiment and interaction on Earth. Again, scientists at CERN do not expect this to happen nor is this something that is being tested.

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Even nowadays , nobody have yet " seen " a single quark



Absolutley correct. Quarks "exist" merely by the indirect evidence of scattering experiments against Hadrons like protons and neutrons. There are in fact other competing theories for what is "inside" these particles but the math behind quarks accurates predicts the "particle zoo" and thus it's the theory that is used today. But you are absolutely correct; unlike other particles, we only know of quarks by indirect evidence.

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About the supposed assumption that a " direct " prove is needed to confirm the theory of the strings
[...] Why should we deny, in priciple, that this can not happen also for the strings ?



I don't and never did. I never said that direct evidence was the only way to confirm a theory... but it is the only way that a theory becomes "solid". As you mention, Quarks are only indirect and thus while popular, there are still people working on alternatives. The majority of the SM is direct and thus there is little to no work on alternatives.

Now, if we had indirect evidence of a phenomena that could only be explained by String Theory, then that would be great! But the fact is that there is NO evidence currently, direct or indirect, and scientists would rather focus on what we can test (SM) rather than what we can't (Strings). But I do admit that because of the high energy regime that is string theory's domain, the first evidence of it's validity will most likely be "indirect".


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