Cold/headache/etc - don't be a bitch, eat like normal. If you want to feel a bit better though then yea chicken noodle soup is great. So is lemon/honey tea etc but they'd make you feel better even if you weren't sick.

Anything involving vomiting - this is tricky. Generally you won't be able to keep much down so there's not a lot of point eating in my experience. More important is to keep sipping water regularly. You'll bring most of it up but at least then you'll have something to bring up. If you're lucky maybe you won't even shock your stomach and you'll keep some of it down and lessen the dehydration you'll have once you get better. If you do want to eat something then avoid acidic stuff. Toast is my general go to when I'm on the recovery road and starting to be able to keep stuff down. Bonus points if there's vegemite on that toast.
Also vomiting pro-tip: If you've got to vomit eat some chocolate. It'll make it taste better provided you're vomiting fairly regularly and it won't have time to sit and go to shit in your stomach.