I think it does and we give our customer's websites a pass-through it. Reason why? Google does not like broken links or broken code. Now that being said, I've noticed that the W3 tool has some issues with tagging of Flash and some Java tags, so take those with a grain of salt.
I am sure it does, when i do my website W3C HTML Validato is 0 errors, then i got better trafic to my website.
It depends. Obviously if you're using incorrect markup, spiders aren't going to be able to tell what is what - e.g. which is the main part of your document (for more emphasis on keywords), what text is relevant to a link and what isn't. For the latter, I've seen too many sites that don't close their links until too late, e.g. there will be an image in the link even though it's not a part of the link. It doesn't necessary cause a design flaw, and many coders - unfortunately - will use whatever code so long as there's not a visibly noticeable difference. Google doesn't see your sight as it appears in a browser. It sees your code. So valid markup is important to an extent. So long as things are where they should be, tags close properly, etc., you're good. Otherwise spiders will have a hard time - again - of telling what is what.