There are plenty of places to wild camp in Ireland, but they're up high on open mountain or in Forestry, or in National parks. As mentioned upthread, all land is owned, but a lot of mountains land is commonage. Coillte, the national forestry company tolerate a small overnight camp in most of their forests, except in Deer stalking season, just after New Year

There's probably not a lot of off road routes in Ireland, We've made public roads of a lot of roads which would remain tracks or trails in other places, but that does mean there are lots of little narow roads (Boreens) to travel through Ireland, many with a strip of vegetation growing in the middle.

Drivers are variable as mentioned. If you were nervous, you could do worse than travel through Northern Ireland, where there is a leap in politeness of drivers, slowing and calmly waiting to safely overtake compared to much of the rest of the place.

There's MTB trails in Ticknock, a few km south of Dublin,  Ballinastoe, near Roundwood in Wicklow, Ballyhoura in Limerick, Derroura in Connemara, Co Galway, and Rostrevor, Co Down
There's unofficial trails around too, use the heatmap feature in strava to see where.

Longer distance off road routes include The 2 Main canals from Dublin, the Royal Canal and the Grand Canal.
The Barrow Navigation from the Grand Canal down the barrow river
The Mullingar Athlone Greenway, 40km smooth tarmac route on an old rail line, from the Royal canal
The Great Western Greenway is 35-40 km on gravel in Mayo

Google isn't too bad for route finding, but it won't route on the off road greenways, and might skip boreens, for bigger&busier roads. I use openstreetmap.org to check and see the latest on roads- its good for checking routes where there's new motorways like the M11 dead end at Rathnew Google shows, or to see if theres a path to avoid a dead end.

There's a lot more road bikes these days as there's a govt scheme to give a tax rebate to purchase a new bike, so I'd imagine you'ld get enough to get a bike shop in most towns with some road bike kit, dunno if you'ld be able to hire them though.

The west of Ireland probably has the best scenery and best places to see, but the least cycling infrastructure
Westport is a great town, Pretty, scenic and good Fun, near the Mayo greenway
Galway is good craic
West clare is pretty and the small towns there are good craic too. Doolin is the only place I know that has camping there

Kerry has several peninsulas, Dingle, the Ring of Kerry, Beara(half in Cork) each of which have places to take your breath away, but they are sharp to milk tourists
West Cork is like Kerry, only less commercial,

Wicklow has loads of forestry roads, and the wicklow way, which you can cycle on a rugged bike most of the way if you are carefull of the walkers, and during the week they' are few and far between.