Genetic variation within a racial group is comparable to genetic variation across groups, making it incoherent as a biological concept even before you get into more-obviously-cultural constructs like the "one-drop rule".
So most serious geneticists have tended to move away from thinking of "race" as closely linked to genetics—the complicated contours of both population genetics and what we mean when we talk about race just don't fit together all that well, and coupled with a history of unethical policies justified by overly simplistic scientific knowledge in this field, it's best to be careful about the terms we use.
Also, what we think of as race typically has a lot to do with cutural & socio-economic traits. Race isn't a clearly defined concept and might mean slightly different things to different people.
Cell also has more than just a single set of DNA - it also inherits mitochondria and maternal mitochondrial DNA.
Maternal phenotype is important only so far as it affects cell development very early in life or has strong effect on epigenetic mechanisms. Those mechanisms are indeed affected by conditions "in the womb".
As for paternal effects, sperm is an antigen and hormone too.
The article seems to be written to a person stuck in very old view of genome and heritability. I'm pretty sure no accrual scientist in the field believes the core thesis.