There is a Thomas Robert Davidson, age 7, listed with his family on the 1880 U. S. Census. The father's name has been interpreted to be Wm C. and N. C. It looks like Wm C to me when it's enlarged. His age is listed as 35 and the mother Melissa is listed as 36. There are four older sisters and one younger brother on this census taken in Lauderdale, AL. On another document Malissa Hendrix is listed as the mother. No D.O.D. found for either.
Family trees on several sites are quite a muddle, not giving sources. I've seen Thomas Robert's name listed on both family trees.
The John Hall Davidson and Katie Baxter Davidson are listed on the 1880 U. S. Census in St. Louis, Mo. She is listed as 18 yrs. old, and John is 24. No child is listed with them. She was born in MO; his birth place has been listed in both KY and LA. No D.O.D. for either was found. She would have been a very young mother if both her and Thomas's birth dates are correct on available records. So, it might be worth checking into the William C. and Malissa Hendrix connection.