I didnt want to go in to this, as it is complicated, but as you are very persistant!
The easiest way to do all of this is to check your fathers title. This should show whether its registered or unregistered. But if you cant or wont, then:
You say you are 90% sure that the property is unregistered. Time to be 100% sure. Get an ordnance survey map of the laneway. Take a photocopy and store the original away safely. Then mark the laneway and take it to the land registry and ask them to do a mapping search. They will tell you if it is unregistered or if its registered, who owns it.
Assuming it IS unregistered, doing a registry of deeds search on the index of names for the builder and previous land owner will only give you a date of a deed- even if you subsequently apply for a copy memorial of that deed, there will be no map attached - whereas a copy of your fathers title would probably contain it.
IF it is both unregistered and doesnt show up on the title deeds, then using the copy map you get a valuation office certificate from the Valuation Office- this will show the owners from 1900 to date. This will be useful if you decide to make an application for first registration based on possession.
Having a copy of the title deeds may show whether there is a ROW on title. However as the laneway has been there since 1958, it is open to anyone to claim they now have a ROw through long user. Even if the laneway only goes to a river/ cliff, is it possible people go there to look at the view/ take a walk/ go fishing/ use it to access their back wall to paint it, maintain it etc etc
You would need an engineer if- it is registered and part of a bigger folio, to mark and certify the area for your application for squatters title. Or if the areas are very small and there is a doubt which folio it belongs to. Or if it is unregistered and you want to make an application for first registration based on possession= again to mark and certify the map.
As I said, if you go to a solicitor for advice they will be able to do all the above for you and advise you appropriately and explain anything you dont understand.