If you’re exporting data from OSM and need to know what’s in the gap then I guess you need to add the highway areas to the map (https://wiki.openstreetmap.org/wiki/Key:area:highway). Mapping the highway area as grass breaks others’ use of the data — for example, someone might use OSM to calculate how much of a town is green space, and having the grass mapped that way would significantly overcount its area.

I’m not entirely sure what you mean about footpaths (if you could link to an example that would be helpful), but it sounds like you’re talking about the links which make the footpath network routable. If the footpath centrelines aren’t joined to the road centrelines at crossings, then routing can’t work. The section of footpath which is on the road area can be tagged with footway=link (or highway=crossing if it’s a designated crossing) to differentiate it from the section above the kerb.