Better not to play "what ifs".  You should at least check on your record and see if the conditions were met.  If they were, the record may already be clean with all charges dropped.

If not, at least you'll then have the information on how much longer to wait, or if you need to request expunging it.  Right now, there are many questions.

Talk to the records division, get your current case disposition, and THEN you can see what options you have.  You might be pleasantly surprised if it's been long enough time and the charges are gone.

Yes . Too, from what I remember it's crimes of physical violence that preclude you from being permitted. Get your record expunged if your cases final disposition allows for it! Don't go for your permit until then, you don't want any type of flags next to your name later if you have to try again. Good luck!

For a handgun, you have to have the specifics on the exact gun you're buying/bought.  For a long gun, the permit is nonspecific and good for as many long guns you want (can afford) in a year before the permit expires.

Unless you have a specific handgun deal you need to pounce on, I'd recommend getting your record expunged first.  If you want to try and see what HPD says now, do the long gun permit first.  That way you don't make the gun's owner wait for who-knows-how-long while you jump through hoops.