When a trust is created, does the actual trust document get filed/registered/recorded with any court or county division which would make it public record?

The answer to your question is "no."  One reason to prepare a trust is to keep the document private.  The only time a trust is filed in court is if someone intends to fight about it.  When a person dies in Florida and they left a trust to distribute their assets, the Trustee is supposed to file a Notice of Trust Administration so that people know there is a trust.  Still, the Trustee does not file the trust itself.