Thank you so much for taking the time to post all that info!
Carter led a large and diverse life. Everything you posted is about his family.
I grew up in the area of NW Illinois where he lived the last 10-15 years of his life. He is a bit of a legend and my grandfather was very interested in him.
He is not acutally buried in a cemetery but on the farm where he used to live. It is located in SEC. 1 of East Grove Township. Apparently, you have to walk in about a 1/4 mile or so from the road (Carter Road) to get to it. It is on private land. My parents went there and took photos when they were young.
I have been enjoying digging up some of the details about his life from old newspapers and so forth. He had a large family in Arkansas, but sadly it appears that all of his children (8 that I know of there) except for 1, died by the time of his death. I do not know yet, if the Margret in the 1870 census is his 2nd wife, or not. He married Ellen Agnes Downey about 1873 and there is no trace of Margret or little Mary.
What do you know about the Nathaniel Carter Williams you are searching for? This one I am working on was married in 1837 in Carroll Co. GA to Martha Avery. He had 2 kids in the 1840 census there and sometime afterwards began the process of moving west. One son was born in Mississippi and then the last bunch in Arkansas where he lived about 20 years. In 1870, wife Martha was still living with a son there. It was said he got in some kind of trouble during the civil war and that was the reason for his move to Illinois. If this sounds familiar, let me know. I do not know who his parents or siblings were.