Hi Peter,
I haven't forgotten you! Just been very busy. When I mapquest 421 E. 16th, where he lived in 1888; it comes up in Brooklyn. There doesn't seem to be and E. 14th St. in Brooklyn though, which is where he lived in 1885; however there are both E. 16th & E. 14th in Manhattan; so I am still going with the belief that he lived in Manhattan. The problem with that is that after abt. 1850 or so, burials were prohibited below 86th St. in Manhattan; and we know he was at least alive until 1888; so he is probably not buried in Manhattan. He could be buried in Brooklyn. Green-Wood Cem in Brooklyn has an online burial inquiry and although about 12 Dempsters are buried there; no Jackson. Here's a link to their site:
http://www.green-wood.com/

He could be in either Cemetery of the Evergreens or Cypress Hills Cem in Brooklyn. You could try writing or calling either of these:
Cemetery of the Evergreens
1629 Bushwick Avenue
Brooklyn, NY 11207-1849
718-455-5300

Cypress Hills Cemetery
833 Jamaica Avenue
Brooklyn, NY 11208-1593
718-277-2900

These are both non-sectarian. Would he have been Catholic?

I wish I could ask you for help in N. Ireland?!?! The only Irish line that I know of which I need to try to trace is of my 4X gr grandfather John Pierce; who I only know was born abt. 1744 in Ireland; somehow ended up in Biddeford, ME USA in time to serve in the Revolutionary War; married there in 1781 and removed to Limerick, Maine. I have no idea what county he came from and asking anyone to look for John Pierce would be like looking for a needle in a haystack!

Lisa