Hello Sue,
Certainly looks like Mary Carabine, the birth you found for John, 1856, again looks correct, you have to remember that its Irish, folks we are talking about its the Nineteenth Century, most ordinary people could neither read or write in Scotland, the register who is writing this down on the certificates is going by the person relaying the information verbally to them with a Irish, dialect, so there would be lots of variations in the spelling.
There is a death for a Mary Carabine Feenie, 1873. age 58 years Holytown, Lanark.
The Parents of Mary Rhind, on the certificate you posted are Patrick Rhind, which is correct going by our research, her mother is Elizabeth, which is correct, her maiden name is difficult to read but it looks to start Br***n, some of our other posters will come on and give a response, however i think its a version of Brannon.
The only thing that i would question is why this lady is mentioned as Ann, and Nancy, on to of the earlier census reports.
Regards,
Jonn.