Cannot find him or family in the 1900 US Census yet.
Sources
"United States Census, 1880," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MD7Q-PZ2 : 14 July 2016), Clay Weddington in household of John P Weddington, District 2, Franklin, Tennessee, United States; citing enumeration district ED 87, sheet 32D, NARA microfilm publication T9 (Washington D.C.: National Archives and Records Administration, n.d.), roll 1254; FHL microfilm 1,255,254.
"Tennessee State Marriage Index, 1780-2002," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:VNH2-GQM : 4 December 2014), Clay Weddington and Fannie Parham, 03 Jul 1892; from "Tennessee State Marriages, 1780-2002," database and images, Ancestry (http://www.ancestry.com : 2008); citing p. 252, Franklin, Tennessee, United States, Tennessee State Library and Archives, Nashville, Tennessee.
"United States Census, 1910," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MG6T-QF8 : accessed 21 February 2017), Clay Weddington, Civil District 2, Franklin, Tennessee, United States; citing enumeration district (ED) ED 21, sheet 9B, family 182, NARA microfilm publication T624 (Washington D.C.: National Archives and Records Administration, 1982), roll 1499; FHL microfilm 1,375,512.
Note, lives right next door to his brother, Thomas, and his wife, Ruth.
"United States Census, 1920," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MNGN-RKK : accessed 21 February 2017), Clay Weddington, Civil District 1, Franklin, Tennessee, United States; citing ED 21, sheet 8A, line 49, family 148, NARA microfilm publication T625 (Washington D.C.: National Archives and Records Administration, 1992), roll 1739; FHL microfilm 1,821,739.
"Tennessee Deaths, 1914-1966," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:NSF3-ZY8 : 13 September 2018), Rufus Clay Weddington, 23 Aug 1927; Death, Franklin, Tennessee, United States, Tennessee State Library and Archives, Nashville.
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DNA Connections
It may be possible to confirm family relationships with Rufus by comparing test results with other carriers of his ancestors' Y-chromosome or mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with Rufus: