A: Some waterways don't have towpaths: the Shannon, the Erne, Lough Neagh, the Corrib, the Slaney, the Shannon-Erne Waterway, the Lower Bann (I think).
Some abandoned waterways have towpaths with cycleways: the Newry Canal.
The three main waterways in the republic have towpaths: the Royal Canal, the Grand Canal (including branches) and the Barrow Navigation. These three waterways are controlled by Waterways Ireland, the navigation authority, and cycling on towpaths is contrary to their byelaws. That said, many people do cycle on them, although that fact would be no defence if you were to cause an accident or injure someone
If you decide to cycle on the towpaths, it would be a good idea to give precedence to walkers, especially children: many stretches of towpath are popular promenades.
The quality of surfaces varies from roadway to rough or boggy grass; mountain-bike rather than racer territory.
The best source of information is John Dunne's book "Towpath Tours: A Guide
to Cycling Ireland's Waterways" ISBN 1-903464-75-7 , published by Collins