Robert Taggart married Margaret Ann Abraham on 12th March 1866 and his occupation is given as gamekeeper, living at Colebrookepark. Colebrookepark is (to this day) Lord Brookborough's home. In 1878, when Robert was born, the place of birth was Corralongford, which is adjacent to Colebrookepark. One of the estate gate lodges is in Corralongford, and Robert senior was still a gamekeeper. So all that information confirms he was a gamekeeper on the Brookborough estate. The owner of the estate during the period Robert worked there was Sir Arthur Douglas Brooke Bt (ie Baronet). So that generation hadn't made it into the nobility. Nobility was conferred in 1952 when Sir Arthur's son, who later became Prime Minister of NI, was made a Viscount (so he was the 1st Lord Brookborough).
The discrepancies in William James date of birth are normal. In the 1800s in Ireland people generally didn't celebrate birthdays and often had no idea of precisely when they were born. If officialdom later asked for a date of birth, it was common just to make one up, with the added consideration that if joining the military, it was sometimes necessary to adjust it to improve your chances of being selected (to avoid being too young or too old depending on the circumstances).
I have looked at the birth records on the GRONI site and can only see births for the 6 children you have mentioned. No sign of William James. There's a small gap in the births between 1868 and 1871, so he might have been born in 1869 or 1870. It's possible his birth wasn't registered. That did happen especially in the early years of birth registration (it started in 1864). You could check church baptism records to see if you can find him there. Colebrooke & Corralongford are in Aghalurcher parish. Church of England in Ireland is the Church of Ireland. There are several churches in the parish. If you don't know which your family attended try Fivemiletown. It is possible the family attended Aghavea, though it is a different parish to where they lived. Copies of all those records, for the years you need, are in PRONI in Belfast. (Personal visit required to view them). Another possibility is that William James was adopted, which could account for the absence of a birth certificate. Formal adoption law wasn't introduced in NI till 1928, so in the 1860s it was a word of mouth arrangement with no records.