Is it more correct to say What happened or What's happened?
Either is correct depending on the sentence:
What happened at the party? What's happened at the party.
What's happened to the cookies? What happened to the cookies.
~Note: This is because the word "what's" is a contraction of "what has", so the sentence really means, "What has happened to the cookies?" That's a different tense than past tense, but either can be used.
Yes! "She is with me and John." You can test it out by using the sentence with just one of the persons at a time. For instance, "She is with me." is correct. Also, "She is with John." is correct. Therefore "She is with me and John" is also correct. (However it might sound more natural to say "She is with John and me")