Were black activists allowed into the major conventions such as the Seneca Falls Convention? Did race even play a part in women's rights?
Yes and no, but mostly no. See there was really no united Women's Rights Movement before the turn of the century and there were various offshoots of activists in various countries at various times. Though there were strains of what we could call proto-feminist or proto-women's rights thought beginning at the turn of the 19th century, these individuals usually hooked onto a particular injustice, usually centered around property or women's ability to own property and testify in court on their own behalf.
With the rise of the abolitionist movement, we begin to see broader enunciation of women's rights, as these antislavery groups realized that the injustices they described (unenfranchisement, sexual exploitation, and economic marginalization) could also apply (to various degrees) to women as well. This is particularly true for William Lloyd Garrison's group and he was a strong (and at times only) proponent of complete equal rights for women.
Within his movement, we see attempts to bring in various northern African American free women to speak on behalf of women's rights, though usually also tied to the topic of slavery. However, these individuals were not as popularly known outside of New England and these brave souls are often driven into obscurity by threats of violence, as were the Grimke sisters (white southern emigres who were strong women's rights and abolitionist proponents).
The Seneca Conference was the brain child of white American women who were rejected from sitting with the men during England's international women's rights conference and Garrison refused to be seated without them. However, this did not mean that Seneca would reach out to all minorities or groups either, and it was largely a white middle and upper class affair. This class component is critical, because the women's rights movement was a highly bourgeois affair and was often contemptuous of women's working-class identities and unique perspective on women's rights (the politics of the mothers vs. daughters, not in the literal sense but in an strategic political sense). This class aspect would remain in place until a few decades after the turn of the 20th century.
E. C. Dubois and others illustrate how women tried various tactics, often using their black sisters, in order to gain suffrage, more property rights, legal recongition, and etc. However, with the failure of many of these strategies, a more racial language was used in order to justify white women getting the vote. This severed in many ways the interracial movement. Add to that the fact that by-and-large, white southern women were largely hostile to black suffrage in general, and you would see the underpinnings of racialized women's rights for the better part of 40 years.
I will say that again, since the women's rights movement had various offshoots, it is hard to portray one solid agenda or strategy. We can say that black activists were involved in some aspects of the movement at various times and places. But due to the Civil War, Reconstruction, and the rise of the Jim Crow era, black activists were largely separated from the white women's rights movement. A fissure that endured even after Second Wave Feminism took hold of the movement.
Sources: Christine Stansell, City of Women: Sex and Class in New York, 1789-1860 (Illinois, 1987); Barbara Welter, “The Cult of True Womanhood, 1820-1860,” American Quarterly (1966); Ellen Carol DuBois, “Outgrowing the Compact of the Fathers: Equal Rights, Woman Suffrage, and the United States Constitution, 1820-1878,” Journal of American History (1987); Mary Ann Clawson, “Looking for Feminism: Racial Dynamics and Generational Investments in the Second Wave,” Feminist Studies (2008); Christine Stansell, The Feminist Promise: 1792 to the Present (2010).
The Seneca Falls Convention didn't occur until 1848. Frederick Douglass was present for the proceedings and said that he, as a black man, could not accept the right to vote as a black man unless women also received that right. He was the only black person at the event. Black women's voting rights weren't on the radar as a possibility in large part to the remaining presence of slavery. Race didn't play a major part in women's rights until the second wave of feminism in the early 1960's.
Another interesting question is whether advocacy for black women's rights was even a thing in the early 1800s.