Blue jays are nasty little customers, no doubt.
They eat just about anything that they can kill without a fight, chase away hawks, crows, cats, and even people, and do a number on wasp nests.
As for that sparrow nest, do the adults have a marking like a black dot or bib on their chest? If they do, then no big loss, they're house sparrows, more properly known as English Sparrows, and they're not native. Do the world a favor and kill the entire nest. Same with starlings, they're also from Europe, and they'll kill bluebirds and other songbirds to take over their nest. They'll even make their own nest right on top of the bodies.