Ok here's my question a few years ago I had gotten tied up with a domestic violence charge that I had no part of falsely accused. Went to court the other party failed to show up and I walked out with a paper saying it was dismissed with out prejudice. I also have 1 speeding ticket and 2 faulty equipment tickets on my driving record. Do you think it's a waste of time to file for the permit. On a side not about a year ago I purchased a hand gun from a dealer and passed the check.
First, how long was "a few years ago"? ... Second, so long as you were not convicted of the DV charge, you will be fine. You may get a call from the AG's office asking you to explain the circumstances. As for moving violations, as long as they are just speeding tickets, taillights out or something along those lines, that will have no bearing on your application. The fact that you were able to pass the NICS check should give you your answer...
If you have any additional questions, or if I can be of any assistance, please feel free to ask! Good Luck!!!