Not that weird in fact. For hereditary information too be properly passed on, the mechanism must be relatively stable and somewhat controllable. The known epigenetic mechanisms (e.g. methylation patterns) fit this bill.
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.