.... If so, she would have married Adam Sutherland under the name of Catherine Ferguson. In that case, both the Sutherland marriage and James Ferguson's death probably occurred some time between 1862 and 1866.
If Catherine had married in Scotland, her marriage registration would have been under her maiden name not a married surname.
Why the window of 1862-66?
It might be an idea for your friend to view Archibald's death in 1855. This would confirm how many children William and Margaret had had by 1855 - boys/girls, alive/deceased.
There is the big possibility that Catherine died between the 1841 census where she shows per Ev's posting and the next census of 1851 where there is no record of her from then on.
Why does your friend think that the 1898 death in the US is hers?
On the facts here, we seem to go from an 1841 census entry to her death in 1898. We do have the 1880 US census entry. The likely birth for the daughter from that entry, Catherine born in England c. 1867 does show on
www.gro.gov.uk I think. This entry shows in West Derby which fits with the possible marriage where that Catherine showed as born in the Liverpool entry. From the GRO, the mother's maiden name is given as Sutherland (without ordering cert, we can't tell if this was an error or correct).
I do not see so far a confirmation that this Catherine Snr had a maiden name of Stirling as is connected to the Kirkintiloch Stirling family of William and Margaret
Monica