I do not have any information on an L Frank Coulter. If you have documentation that he was born in Rutherford County he could be the son of Anderson Coulter or one of the other Coulters who show up in Rutherford County records. I haven’t found documentation on his children since Anderson and his wife died before 1850.
I have census records for an L F Coulter in Fayette County, TN 1850 census in the W B Stafford household; 1860 and 1870 census of Shelby County, TN with wife E Fannie Coulter. In 1860 he is in the household of J A Coulter (?brother, cousin). 1830 Census of Fayette has a John Coulter, this may be a lead for you. I haven’t looked at the court records in Fayette and Shelby Counties. Ashley and Allen Coulter son’s of Henson Coulter were living in Hardeman County in the 1830’s and 1840, which joins Fayette Co.
Good Luck,
Elaine
tnelaine@bellsouth.net