For someone who has a lawyer, you have an awful lot of questions. Did you discuss these questions with your lawyer before you plead guilty? That was the time to ask them. I will try to address your questions: If you have to do community service, that means you received a Conditional Discharge and therefore the case does not seal for 1 year. During that time, a background check may show the arrest charge and what you plead guilty to. After that time, a background check by law enforcement or an educational institution (I think) will show the case and outcome. You do NOT have a criminal record as a result of this plea as disorderly conduct is a violation and not a crime. If you are asked if you have been arrested, the truthful answer is yes.
The above is for informational purposes only and not meant as legal advice.
First off, you should talk to your lawyer. I am not sure that I understand your question and he is in the best position to help you get what you want or to understand what you have done. Did you plea to a class B misdemeanor? If so, that will not be sealed and the arrest for falsifying business records will always appear on your record.
If it was a disorderly conduct violation, the arrest for the 240.20 misdemeanor will still show unless your attorney moves to seal it. However, you will always be able to claim that you have no criminal record. In New York, a violation is not a crime.