Does the Nielsen-Ninomiya theorem apply to quarks? Since quarks are massless before EWSB and left & right quarks have different weak hypercharge, there should be additional species of quarks when it is put on to a lattice, correct?
Yes, it does apply. Yes, there are additional species of quark "doublers" in lattice calculations. (The latter statement holds regardless of zero vs. non-zero quark mass.) Even though Nielsen and Ninomiya put the word "neutrino" into the titles of a couple of their papers, the Wikipedia article linked in the question correctly states that their no-go theorem applies to lattice-regularized "fermions" more generally.
Any introduction to lattice gauge theory should provide many details about this issue and the main techniques that have been developed to address it over the past ~35 years (rooted staggered fermions, Wilson fermions, domain-wall fermions, overlap fermions). If you don't have easy access to any of the textbooks on the subject, Rajan Gupta's 1998 Les Houches lecture notes should still be fine.