Sorry I can't be of more help re: Sullivan & Tarrant. It isn't my direct line & the lady who could have helped, Lillian Weiss, is now deceased . She knew so much about my family, the Kelly's, the Moran's, the Tarrants etc. Have you looked at Holy Name Cemetary in JC to see who is buried in either Tarrant or Sullivan graves? Or check with Hudson Co Genealogy group whose volunteers transcribed many of the names in Holy Name which gives date of burial & grave & age of deceased. You can call them & they may help over the phone. They used to but it has been years since I have called. If you know the DOD & they have the burial they can tell you who else is buried in the graves with names DOD & age. You can also rent the film from LDS FHL. I don't know where you live but you can see if there is a center near you. They may have the film on site, but you need to check.
I can't help you with the horseshoe section downtown. Spruce St is in Jersey City Heights section, as is St John The Baptist.