Dear Anne
Welcome to Ireland Reaching Out
They are indeed interesting names! Unfortunately, many of the Church of Ireland records for Cork City are lost. However, you could search the sites suggested below for records of the Fleming family in Holy Trinity, St.Luke's, St. Mary's and St. Nicholas Parishes:
www.familysearch.org has a huge database of genealogical records including some church records for Ireland.
http://churchrecords.irishgenealogy.ie/churchrecords/ have begun a project to upload Irish church records to their site free of charge. However, so far only Counties Dublin, Kerry, Carlow and parts of Cork have been covered.
Other important sources include the Tithe Applotment Books and Griffith?s Valuation:
Tithe Applotment Books (1823-1837): http://titheapplotmentbooks.nationalarchives.ie/search/tab/index.jsp
Griffith?s Valuation (1847-1864): http://www.askaboutireland.ie/griffith-valuation/
Some other websites that may be helpful are:
The National Archives of Ireland: http://www.census.nationalarchives.ie/search/ you can search the 1901/1911 Census on the National Archives page too
Slater's Commercial Directory 1846: http://www.failteromhat.com/slater.htm
Cork Archive Institute may have papers in relation to the business, or at least mention of it:
http://www.corkarchives.ie/
Remember to post any new information that you find here. The more information you post, the more likely it is that one of our volunteers will be able to advise or assist you. Also include information concerning which sources you may have already used so others may further your search.
Best of luck with your search
Clare Doyle
Genealogy Support