My App uses openstreetmap files to show waypoint (or coordinates) on a map.
Based on an Openstreet map file: how can I determine the shortest distance from a GPS coordinate to the (walking) path.
Can I read an OSM (map) or .osm file ... to access the paths?
There are some misconceptions. First, the OSM file is not the map file. The OSM file can be used to create a format to make map rendering fast e.g. mapsforge format does this.
With e.g. GraphHopper you import the OSM file like .pbf or .osm which then creates a graph which can be used for problems you described. Not exactly sure what problem you have though ;)