You might want to talk to an attorney familar with the process, but keep in mind if this is an administrative proceeding then the rules are a lot looser and the fact that you were not convicted of many of the things on your list of former charges has little to no bearing.
Frankly you would have a hard time doing many things in life, getting a security clearance, getting a medical license, FFL or real estate license for that matter....
In all honesty with a list like you have I'm not sure I feel bad that you are not getting state permission to carry a gun around. For one thing few people have the arrests you have even if you have managed to get many charges dismissed.
Furthermore I call attention to 2 things, 2 of your charges are for making out in a park, and loud radio.... now in my experince working closely with LE in a number of jurisdictions across a few states, it's rare for any officer to want to write paper on offenses such as this unless the individual is really a problem.... or the officer has some sort of issue.... so either you have very bad luck or your interactions in public tag you as someone LEO's feel "needs to be taught a lesson".
Think long and hard about this, maybe that was you then but is not the case now, great is so I congradulate you but it's going to take some time to sort through this, prove what your are here and now and maybe being in a state friendlier then NY.