First, congratulations on your current path of life and recovery!
My experience says medical school is a possibility in regard to the drug charge, however, the assault may give you more trouble. I personally had 15 years of addiction including two misdemeanor possession charges for heroin (2012, 2013), one for cocaine (2007), and larceny under $1000 (2013-stole from Walmart) - I'm just about to wrap up my first year of medical school (yes a US medical school). That being said, I had 7 years between my addiction, criminal history, and my application to medical school along with an extensively documented history of long-term residential treatment, being an impactful and productive citizen, documented sobriety, and a high level of performance during undergraduate studies.
My interview was fairly easy for me - meaning I didn't have to answer the cliche interview questions because my criminal history and charges took up the whole time, but we had a good time just talking about it, what it was like, and what I've learned through the whole experience.
My personal opinion says you should plan to take a gap year. I'm not sure what your stats are like, but even with a stellar cGPA, sGPA, MCAT score, and ECs, you face the disadvantage of not having a lot of time between your last criminal charge and where you are today, a violent offense regardless of the circumstances unfortunately, and some medical concerns.
Lastly, and this is a common issue with the recovery community, but I am unsure as to how you define "clean and sober." Do you consider complete abstinence sober, or as long as you're not doing heroin you can still drink? The question isn't an indictment on your personal views, and what works for one doesn't work for others although the data and my personal beliefs say complete abstinence is the way to go, but if you do use alcohol I would consider what your social media platforms look like because ADCOMs would probably view alcohol use as a high risk factor for relapse in considering your application.
Hope that helps and feel free to ask me any questions - best of luck