The "crown copyright and database right 2015" message on the map is slightly misleading. Certain OS mapping has, since 2010 been available under an open licence. In February 2015, they moved from an OS-specific open licence, to OGLv3, but for unknown reasons they continue to want the attribution as shown, with no reference to the OGL. The map pictured is one of the OGL licenced products, data from which can therefore be freely included in OSM, as with the OS imagery layers in the iD editor.

The paths Fife Council added to the map would default to being copyright Fife Council, since they don't appear to be based on features on the OS open map, but since there's no attribution for these I would just go ahead and assume they can be freely added to OSM.