Dear Bertrand,
Here are my answers to your questions:
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A similar question was asked and answered in the following forum topic: The frequency component of flapwise - #4 by Jason.Jonkman. The new linearization functionality for floating offshore wind turbines mentioned in the post dated Nov 02, 2018 has now been submitted as a pull request into OpenFAST–see: github.com/OpenFAST/openfast/pull/350.
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Correct, but there is only one static-equilibrium position in the absence of external loading from wind and waves, so, only one simulation is required.
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Correct. It is always best to perform an eigenanalysis about an equilibrium position, so, load case 1.2 is really a prerequisite of load case 1.1.
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The difference is regarding which outputs are being reported for the code-to-code comparisons.
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The linearized natural damping can be derived directly from the linearization analysis / eigenanalysis. Free-decay tests are often used to derive the linear and quadratic (viscous) damping (via the so-called p-q analysis of the free-decay time series response).
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No, these natural frequencies do not look correct (assuming you are reporting values in Hz). The natural frequencies of the OC4-DeepCwind semisubmersible are reported in Figure 3 of the paper you attached.
Best regards,