You punched someone not once or twice, but three times in the face because they grabbed your flashlight? Yeah....I'm not sure Fire/EMS is necessarily the best career option for you, considering most fire departments are very paramilitary run, and quite frankly something tells me that there was more than this one incident during your <10 week stretch in the Army, because they booted you out instead of merely an Article 15. Perhaps an anger management course instead?
As far as background checks it depends. For a small mom and pop dialysis derby BLS company you might just get fingerprinted and compared to the criminal database. For a big city fire department that employs background investigators, you better believe that'll come up easy peasy.
And if it is discovered via any means, and you omitted or lied about it, yeah you'll get disqualified fast enough to make your head spin. Besides, most applications ask you for your employment and education history's and account for any periods of unemployment, how exactly are you going to account for that 2-3 month gap that isn't either?
Your best bet now is time and distance. Go to school, get a degree. Get and hold a job awhile in school. Stay out of trouble. Remember you already have an uphill climb compared to other applicants, every single speeding or parking ticket just adds weight to your rucksack, and anything worse than that......and take an honest introspective look at your life. I don't care what you say to your family/friends or anonymous strangers on an Internet forum, just be honest with yourself, why did you snap like that in Basic? Everyone else managed the stresses fine, why not you? What happens when it's 3am, its been a long day of drills and being corrected by your captain on what feels like every detail of all your tasks that day and your pissed and tired and now you're the rookie in the back of the ambulance alone with a patient who for whatever reason isn't cooperative, maybe feeling a little froggy, maybe they've said something to you already and now when you go to put the blood pressure cuff on, they grab it from you instead, just like that flashlight....what are you going to do? If there's any possibility in your mind that you may hit them, then a) go get anger management and/or impulse control counseling now before you find yourself being ordered to do so, and b) this really isn't the best career field for you at all.