Comic Strips

Incredibly generally: gender is the expression of gender identity and is a social construct while gender identity seems to be largely influenced by biological factors. Sex is the biological differentiation, and while the delineation between the sexes is culturally defined (if someone has xxy sex chromosomes, high testosterone, a penis, and a vagina it's a cultural decision if we say they're male, female or intersex), it's a classification based much more on observable factors.

Race and ethnicity are more akin to sex than to gender identity, which would be better compared to cultural identity.
What distinguishes races is a social construct, but within a context racial classifications are relatively consistent. Racial markers that mean nothing in the US might be quite significant in Rawanda.
Similarly ethnicity, being a blend of race, language, culture and heritage is socially constructed but relatively objective within a context.
Culture on the other hand is, like gender identity, more to do with subjective feelings, opinions, and choices on the part of the individual, with the distinctions between them being cultural.

The woman in question mislead people about her race and ethnicity by misidentifying her relatives and heritage. Her cultural affiliation is harder to dispute, although being a chapter president for the NAACP shows at least a degree of acceptance by the African American culture in the area.