Well, they mention hydroponic growing so that removes the soil issue altogether.
I believe if you fired nuclear waste into the sun you would miss, but if you DID do the math right and managed to get in on course for the sun, it would probably burn up long before hitting the actual gass explosions the sun is made of. While this might ever so slightly affect the suns lifespan, I couldnt say wether it would be a postive or negative effect. I think it would depend on the efficiency of the cells the material was used in: anything left that is fissionable would ignite and create heat and light, any material that has had almost all potential energy removed would sap light and heat from the sun, but without a rate of transfer I really have no idea what the effect would be over time.
As a one off event, it would do very little and would destroy the material forever, unless you had collected such a huge mass of waste material that it managed to exhaust all energy the sun was outputting, to the point that it had not enough left to continue fission and release more. But for that we would have to have found a planet of ridiculous proportions made entirely of unstable reactive materials, then mined it, then expended the potential energy by transferring it to our storage grids.
BAHJURD: TL;DR version is... Fuck all. We wouldnt even notice an effect, it would simply vanish in space.
