Question #10 reads "Have you ever appeared in court as a defendant for any criminal offense (excluding non-criminal traffic offenses)?"
No matter how you slice it, you cannot plead no contest unless you are a defendant to a criminal offense. So, right there, you have to answer a question in the affirmative that will result in your providing the same explanation as a yes answer to any of the questions about convictions.
I expect that one of the lawyers on this list will clarify that a no-contest plea is indeed a conviction. Furthermore, the licensing officer (who will almost certainly find your record of the no-contest plea) will not be amused by creative word games that are variations on "I was never convicted of a felony since I plead guilty."
If you have anything other than a completely spotless background, you should be consulting an attorney, not the forums, before submitting your application as you have once chance to get it right.