My research on the Thomas Rogers' line has been basically trying to untangle the cross information that has happened between Thomas Rogers' son, Thomas, and my James Rogers' son, Thomas.Both have been listed as having the same birth date and death date on several sites which is inaccurate.
In terms of George Rogers and family, I did find George in the 1820 census in Unity twp., with John listed next to him and Alexander nearby. However, they appear to be in a section on the very west side of Unity twp. so it is possible the person taking the census wandered over the line into Fairfield twp and counted them in the Unity census. Or perhaps Geroge's land was in both townships.
I also found a John A. Rogers, son of George A. Rogers and Alice Fife born in Fayette Co. PA in 1792 and who died 1 Sept. 1851 in Unity Twp. and was buried in Kemble Cemetery, in Elkrun Twp., Columbiana Co. Ohio.John had been a private in the Army in the 1812 War and served under Captain. Isaac Warner's Co. from March 11,1812 to 11 Sept. 1814.John married Phebe Kemble in 1814. The source: OHIO WAR of 1812 SOLDIERS' FAMILY GROUPS.
That is all I have on George and his family at this point, I hope it helps.