No, I am using your notation and don't use the K3 double cover at all.

I would use "free product with amalgamation" only when the upper corner group injects into the other two groups. This is also the case when you have a really nice description of the result. Otherwise you only get a presentation (given presentation for the three groups) and we all know how difficult such can be to handle...

This has nothing to do with $H$ being a Hopf algebra but is true for any semi-simple algebra (over say an algebraically closed field so there are no division algebras involved) and follows directly from the Wedderburn classification of semi-simple algebras.

I don't understand the second paragraph. Finding such an $m_\alpha$ should mean giving a section over $U_\alpha$ of the canonical double cover. However, that is not possible on a Zariski open $U_\alpha$ as then the canonical double cover would be trivial.

As far as I can see simple modules are finite dimensional.

This of course is a general fact for algebraic groups and there are references for that. However, a reference for abelian varieties is given by David Mumford: Abelian varieties, Ch 4 (iii)