I don't think a late application hurts, granted I didn't finally decide on applying to med schools until the end of summer, so I submitted my application really late. I think I was complete early to end of october for most schools. Fortunately I still received 15 interview invites so far, with half of them from top-10 schools, and I already received a few acceptances. I haven't written any updates or letters of interest. I asked my professor about them and he said it really depends on the school, most accept them, but there are some that don't read them until post-interview. I don't think a late application is necessarily a death sentence-it probably everything depends on the strength of the app as a whole though.
What will help you is to have someone submit a letter of recommendation for you, preferably from someone who knows you really well and can vouch for you as a good candidate. Good luck.