The term unadopted road is generic - it just means that the local authority do not maintain it. It may still be part of a public right of way. There are various types of public rights of way - carriageway, along which vehicles may travel, bridleway along which horses and bicycles may travel, footpath - which is pretty obvious.
It is very possible that this is part of the public carriageway - it just isn't made up.
There are plenty of public highways that don't lead anyway - cul-de-sacs.
As far as I know, if there is a public right of way along there, the 'rules' about minimum distances from structures apply - you could check this at the local Planning Department. They will advise you as to whether you need to apply for PP. Apart from the hundred and odd pounds fee, its no big deal - you will very likely get it. Alternatively you could just ignore it - many folks do just that - and erect the summerhouse. The worst that can happen is you are asked to remove it.