Rebecca's birth date must be estimated based on census information and her marriage date. She and Joseph married in August 1840, and the 1850 census lists her as 22 years of age, meaning she would have been 12 at marriage, and the 1860 census lists her as 35, meaning a marriage at 15. It was not unusual for young females to marry at 15 while 12 seems incredibly young.
Rebecca died before 6 October 1867 when Joseph married Emmaline Patterson in Ripley County, MO.
I have not verified the following, which comes from a descendant of the family.
Children
I have not additional data on Lavina, Manerva, Richard, Davis, or Caroline.
The only additional information I have on Erwin/Irwin is that he was in the family of his brother James Calvin Smith and wife Nancy Catherine Walker in Oregon County, MO, in 1870. See link for James Calvin.
Joseph Smith to Rebecca Lax, 24 August 1840, Henry County, No Return
I believe this was the data the application to marry was made. The MG, JP, or other official performing the marriage never turned in the reord.
HENRY COUNTY TENNESSEE MARRIAGES VOL I 1838-1852, compiled by W. O. Inman from Marriage Book A Winchester Printing Co., Murray, KY, 1974]
Joseph married Emaline E. Patterson 6 Oct 1867 in Ripley County MO.
NOTE: I'm uncertain whether Patterson was a maiden name or the surname of a deceased husband. She may have been the Emaline E. Barker who was married to James Patterson, but I'm only speculating. No records support my hypothesis.
1850 Henry County, TN, p409b Civil Dist 18, #58: Joseph Smith, 35, Rebecca, 22, Viney Ann, 9, Catherine, 7, Minerva, 5, Robert, 3, Richard, 11mo, Rebecca born NC, all others TN. Robert Clinard, 25, lived with them.
1860 Henry County, TN, p495a/b: Joseph Smith, 44, Rebecca, 35, Livinia A, 20; Catherine, 16; Manerva A. or H., 14 (most likely A.); Robert H., 12; Richard, 9; Davis, 7; Calvin, 5; Carroline, 3; Joseph T., 1; all children born in TN
1870 Ripley Co., MO, p667b: Joseph, 50, Emeline, 32, Calvin, 17, Joseph, 11, Erwin, 9, all born TN
Separated by one family: E. D. Smith, 43, Lucinda, 37, J. F., 16, Melissa, 14, Martha, 12, N. M., 10, W.(?) L., 8, Malinda, 4, Sarah, 1
I'm far from certain this is our Joseph; the name is too common, and there was no John listed with Joseph and Emeline in 1870. They were in the same county as Calvin and Irwin, and the age corresponds with our Joseph's 1850 age.
1880 Oregon Co., MO, p371: Joseph, 60, TN, and John, 18, son, MO, parents TN: