Getting into NYU from small school

I am currently ending my sophomore year and I plan on applying hopefully to NYU,Columbia,Uni of Miami, and Notre Dame. My school offers zero AP classes but I am taking the highest possible classes offered. Would no APs and no class rank hurt my chances. My school is to small for class rank with a sophomore class of 14
I have a 4.14 GPA and am currently taking Chemistry honors,Western Civilization Honors, Humanities English Honors, Latin 3,Theology,Rhetoric, and Pre Cal B
My extracurriculars are softball, strings club ( play the cello ), Red Cross club, National Honors Society,and I volunteer at my local hospital
SAT-2000 planning on taking it 2 more times Junior year

Are NYU and Columbia out of the question?