Typically, your common white egglayers are Mediterranean breeds.
However, Barred Hollands, a dual purpose breed, lay white eggs as well. There are other misc. breeds, such as Sultan, Polish, and Spitzhauben that lay white eggs.
A bird that is black with white spots on its wings might include the ivory billed woodpecker or the jackdaw. This might also be the tri-colored black bird.
brown and white
They are called White-Spotted Brown spiders
A white-breasted Nuthatch might be the bird you're looking for.
White with brown spots! http://www.lindahall.org/events_exhib/exhibit/exhibits/womenswork/price1.shtml
A killdeer lays these eggs but they can sometimes be blue with brown spots
Sounds like a killdeer.
Sounds like a humming bird.
A garden orb weaver probably
It could be a number of birds (a picture would have been nice... or some description of the size... or something other than it's coloring..-__-) So I'll shoot in the dark here: Golden-bellied Fly Catcher Some type of Warbler Wren Or the American Gold Finch. Maybe I helped with your vague description of what MIGHT be a bird?
a coral
The insect that can fly and has white spots on its wings is a ladybug.