Hi, I have found some info which should help you in your search.
1901 census:
Captain St Lower,Coleraine,co Derry
William Doherty age 60,head of house,roman catholic,born co Derry,farmer,married
Jane Doherty age 46,wife,roman catholic,born co Derry,married
Mary Doherty age 22,daughter,roman catholic,born co Derry,single
James Doherty age 17,son,roman catholic,born co Derry,single
Agnes Doherty age 5,daughter,roman catholic,born co Derry,single
*this is your grandfather and his family
1901 Captain Street Lower,Coleraine,co Derry
Grace McLaughlin age 67,head of house,roman catholic,born co Derry,widow,spirit dealer
Jane Doherty age 12,niece,roman catholic,born co Derry,single
Grace Doherty age 9,niece,roman catholic,born co Derry,single
*this two young girls are also your grandfather's sisters
1911 census Captain St Lower,Coleraine,co Derry
William Doherty age 71,head of house,roman catholic,born co Derry,married,spirit merchant
Jane Doherty age 60,wife,roman catholic,born co Derry,married 34 yrs,10 children,4 are living
James Joseph age 26,son,roman catholic,born co Derry,single,labourer
Grace Doherty age 18,daughter,roman catholic,born co Derry,single,scholar
Agnes Doherty age 16,daughter,roman catholic,born co Derry,single,shop asst
Your grandfather's parents William Doherty and Jane McLaughlin were married in 1875 in co Derry
your grandfather and Annie Mulhern were married in 1921 in co Derry
I can explain how William became a publican. Grace McLaughlin's husband Edward McLaughlin was a publican who died 10 August 1893. He left his estate to his wife Grace. Upon her death on 7 March 1904 she left her estate to William Doherty which is how he went from being a farmer to a spirit merchant.
children of marriage of William & Jane:
Mary born 7 August 1877 Drumachose,co Derry
John born 29 June 1879 Drumachose,co Derry
Margaret Bridget born 1881 co Derry
James Joseph born 1884 co Derry
Edward born 1886 co Derry
Catherine born 1887 co Derry
Jane born 1889 co Derry
Grace(Margaret) born 1892 co Derry
Agnes born 1894 co Derry
George born 1896 co Derry
I found a birth I believe to be Annie's. It is Annie Huey Mulhern born April-June 1893.
If you wish to see the full childrens' births,Annie & James' marriage or the marriage of William & Jane all are on the rootsireland.ie website.You have to buy credits (as it is pay per view) if you wish to see entire cert or you can order the certs online from GRONI for 4 euro each which may be a good idea as the names of William & Jane's and Annie Mulherns' parents may be on them,which will bring you another generation back.Hope this helps,Kim