hmm, you should say "What is happiness?" (it seems more serious), not "what's happiness?" (it seems you don't know the word's meaning) And you should say "what is happiness for you?" or "What does the word 'happiness' bring to mind?" ('what does... mean to you?' is for something you have already, eg: my family means a lot to me.) :)

Yes, but in the right contexts. "What's happiness...?" is very informal (e.g. text message to friends/family). "What does happiness...?" would be used in formal or slightly informal contexts (including high school assignments!).