It's up to you whether you peel or don't peel your zoodles! The peel adds some extra green which is awfully pretty but it's really a matter of personal preference.
Can you leave your zoodles raw? Sure you can.
Chicken should be cooked but doesn't necessarily need to be warm when you combine the ingredients. This is a cold dish.
I don't recommend making this dish in advance just because of the nature of zoodles. They can get mushy so I like to enjoy them right away.
If you do make this in advance or have leftovers store them in an airtight container in the refrigerator. I'd eat it within the next 24 hours just to avoid overly soft zoodles. When you're ready to eat do not reheat. Serve cold.