First let's corrects things. 1+ is grammatically correct, but doesn't mean the same thing as 1, and 2+ is not correct at all. I'll start by the second one.
"La passion fait ressortir ce qu'il y a de meilleur en nous" feels very easy to translate : "Passion brings out what's best in us". It seems very close to me, can you see the similarity ? Literally it's "The best there is (in us)" or "What there is best".
The first sentence is based on the same principle, but with an "extra layer". I don't know if there is a equivalent in English, but you can break it down like this : First consider "ce qu'il y a d’exagéré en nous". It's the same construction, except the adjective is different. It doesn't have to be a superlative : "Ce qu'il y a de bien avec cette bibliothèque, c'est qu'on est sûr d'avoir de la place".
And then "ce qu'il y a" becomes "ce qu'il peut y avoir" to add a nuance, that could be translated by "can" or "might". There might be something exaggerated in the expression we're using.
Don't hesitate to ask me for more clarification, the post was already long so I didn't translate everything.
And for the disambiguation between 1 and 1+ : There is no way to tell the difference in English if I'm not mistaken, it's quite subtle :
1 is "what could be exaggerated" as in "it might be exaggerated already, it's hard to tell", while 1+ is "what can be exaggerated" but as in it's not exaggerated yet, but is often subject to exaggeration or it's "exaggeratable". If the last part isn't clear don't worry, you won't encounter the second form. Again feel free to ask for clarification if need be.