Some simple questions, for which I know no precise reference (and would be deeply grateful for a nice one!):

Is it true that the category of (pure) polarized Hodge structures is abelian semi-simple, whereas the whole category of pure Hodge structures is not?

Should one only consider those morphisms of polarized Hodge structures that respect polarizations in order to obtain an abelian category?

Is it true that all pure Hodge structures 'that come from geometry' (for example, the graded pieces of the weight filtration of the singular cohomology of varieties and motives) are polarized?