Bud Jr.'s name is correct but the birth index has Leo wrong. While he may have been born Leo Richard Everett, the birth record for his daughter (Thelma Lea) identifies him as Leo Rachel. His middle name is also Rachel on his death certificate of 1956.
Bud Jr was born Sep 1931 and died Dec 1931.
I cannot find anything on Thelma. It is likely that her mother (Joan Esther Loosier) May have remarried after Leo's death and Thelma may have been adopted by the step-father.
Now .... regarding the children of J. Dan and Minnie Joslin-Everett ... I have 12 children for them:
(1) Berry Burleson Everett (never married)
(2) H. John Everett (have no info)
(3) William James Everett Sr. m. Barbara Boyd
(4) Ed Cerry Everett m. Frances Fornett Grubbs
(5) Frank Elmer Everett Sr m. Johnnie Beatrice Carver; Rose ?
(6) Carrie Gertie Everett m. James Edward Carver
(7) Mosell Everett (wife unknown; divorced before 1974)
(8) Odis Robert Everett m. Wanda Pearl Jackson; Nancy N. ?
(9) Nora Jane Everett m. Robert Dewitt Lay Sr.; Milton Reeves
(10) Robert Troy Everett m. Delia Mae Helton
(11) Stella Mae Everett m. Zeezrom Zenos Langston
(12) Tally Mounty Everett m. Carolyn Joann Bentz
I cannot find the h/h of J. Dan and Minnie in 1930.