Like mentioned, ESA's don't fall under the same legal protections as service animals do. So the answer to your question is going to vary based on where you are.
For school, ask your administration and plead your case and see if they will be accommodating.
As far as public places, you can bring your dragon with you anywhere in public. But once you enter private property (despite whether or not it's open to the public) you will be subject to the rules of the property owner. For instance, Walmart would be considered open to the public, but it is not public, it's private property and Walmart would have every right to refuse you entry or ask you to leave if they wanted to.
-Brandon
