Elizabeth Virginia Falkner

Not much is known about my third great grandmother Elizabeth Falkner.  I have not found any evidence of marriage and have been told by family members, she was in fact never married, but raised seven or more children.  While looking at the pieces of accumulated information it is hard to distinguish what is correct. Elizabeth was born in Sandy Creek township, Alert, Franklin Co., North Carolina on 20 Dec1833 as found in the Falkner family bible passed down to a first cousin.  The information was hand written and hard to read. She died 27 Nov 1912, in the same community. Evidence of this was determined by the tombstone placed on her grave in the Faulkner Family Cemetery and her death certificate, which I found on ancestry.com.
Looking at other documents, such as census reports can be very confusing, but with investigation of the documents as a whole can lead you to the truth.  However, the more I look at census reports for Elizabeth Falkner, the more confusing they become.  Here is what I found:

1860 census
Jinny Falkner  age 65  (maybe Elizabeth's mother)
Elizabeth Falkner age 25
James Mosley age 14
Washington Mosley  age 7
William Mosley  age 3
Lucy Mosley  age 6
Mary Mosley  age 6 months
Jane Mosley age 22

1870 census
Elizabeth Falkner age 36
Washington Falkner age 15 (name changed since 1860)
John H Falkner age 18 (not on 1860 census)
William Falkner age 14 (name changed since 1860)
Charles Falkner age 8
Ernest Falkner age 5
Sallie Falkner age 1

1880
Elizabeth Falkner 47
Ernest Falkner 15
Sally Falkner 12
Gertrude Falkner 9

1900
William H Falkner age 43 (listed as head of house)
Elizabeth Falkner age 66

Questions:
Are the Mosley children in the 1860 report Elizabeth's?  James is probably not her child because of his age.
Where does Jane Mosley fit in this report?  Elizabeth did have a sister named Jane Malon Falkner born in 1838.
Which name is correct for Washington and William?  Is it Falkner or Mosley?

Theories:
1.  Elizabeth was never married and all of her children were illegitimate. James in 1860 was her stepson.
2.  Elizabeth had a sister that married a Mosley and when the sister and husband died unexpectedly, Elizabeth stepped in to take care of the children and eventually change the name from Mosley to Falkner.
3.  Elizabeth was a kind-hearted young woman and dedicated to raise these orphaned children.

Where does this leave me? I'm not quite sure.  Am I a Faulkner or a Mosley?