The Hilbert-Schmidt Independence Criterion (HSIC) is a dependence measure based on reproducing kernel Hilbert spaces. This measure can be used for the global sensitivity analysis of numerical simulators whose objective is to identify the most influential inputs on the output(s) of the code. For this purpose, HSIC-based sensitivity measures and independence tests can be used for the ranking and the screening of inputs, respectively. In this framework, our work proposes several improvements in the use of HSIC to increase their application spectrum and make the associated independence tests more powerful. First, we introduce a new method to perform the tests in a non-asymptotic framework. This method is much less CPU-time expensive than the one based on permutation, while remaining as efficient. Then, the use of HSIC-based independence tests is extended to the case of some space-filling designs, where the independent and identically distributed condition of the observations is lifted. For this, a new procedure based on conditional randomization test is used. In addition, we also propose a more powerful test that relies on a well-chosen parameterization of the HSIC statistics: the kernel bandwidth parameter is optimized instead of the standard choices. Numerical studies are performed to assess the efficiency of these procedures and compare it to existing tests in the literature. Finally, HSIC-based indices for functional outputs are defined: they rely on appropriate and relevant kernels for this type of data. Illustrations are provided on temporal outputs of an analytical function and a compartmental epidemiological model.