Free BMD gives a marriage of
Robert William Bagg to Mary Jane Barber Q3 1875, Newcastle
So your Mary Ann was only about 2 when her Mam married, she would probably have been known as Mary Ann Bagg as she grew up, but may have been aware that her birth was registered as Barber.
As for the Carruthers children's birth registrations and the two different MMNs used, it all depends on who is giving the information.
If it was Mary Ann herself, she may have used her birth surname, her husband or another person may just not have thought/known to do that and gave the name by which she had been known before her marriage. Not necessarily fibbing as such, just didn't think about it.
Unfortunately in order to prove that theory one way or the other you'd have to buy the certs for all the children to see who registered the births - which would be really expensive and the theory may not even be right :-)
Yes he did have those two sisters. I did find him named on someones family tree on Ancestry that has all of his siblings described as half brothers and sisters.
If that tree is correct it also means that his mother was just 16 when she had her first daughter (who happens to be my great grandmother) in December 1889. That could perhaps go some way to explain why she used her mothers married name until she was a bit older?
Michael