Has science solved the problem of why something exists rather than nothing?
1) Gottfried Leibniz famously asked why is there something rather than nothing. I know that certain scientists have tried to answer this metaphysical question, although I'm not sure that science can satisfactorily reply to Leibniz. I respect the place of science in rational discussions. However, why something exists when it's possible that nothing might have existed is a question that seems to exceed the purview of science. We must also remember that "nothing" is being used by Leibniz in a metaphysical rather than scientific sense.
2) Is light-speed still the cosmic speed limit? The notion of things going faster than light seems to have been refuted for now. And the second video link you included is less than clear about what "nothing" means. In other words, the term "nothing" can be defined within a quantum context or it can be fleshed out metaphysically. To make the discussion fair, a term needs to be used monosemically as opposed to being used equivocally.
3) I personally do not believe that free will is an illusion. It's somewhat of a mystery how free will exists, but one could argue for free will by appealing to moral responsibility and counterfactual freedom. Peter van Inwagen has written extensively on incompatibilist free will. Nancey Murphy also provides evidence that free will may possibly arise from an initially deterministic system. Think about the robot in "I, Robot" that learns how to wink; maybe free volition can be produced in a similar way from a system (the brain) that's supposed to be wholly deterministic.