I am familiar with some of the Eatons you mentioned, but I don't know the answer to either of those questons. I have the info on Anthony Smith but have never been able to determine who Anne's parents were. She was born in Woodford about 1787 based on the census records. The Eatons of Woodford were non conformists and I haven't found the church they belonged to or any record of them. They were buried in the parish church of Woodford, but since they were non conformists not much is recorded about them. I believe they were Baptists.
There was an Anne Eaton, daughter of Peter Eaton born in 1783. That family lived in Old. That is the closest date that I have.
I am sorry. There just isn't enough info to link them to a particular family. I have been able to do that with Thomas Eaton through his wills. I have looked at most of the wills but don't remember seeing anything about Ann being the wife of Anthony Smith.
Wish I could be of more help.Karen