can gpx files be generated in OSM? If not, how can I do this?
Various options are available - what do you actually want?
Gpx files that have been uploaded to osm can be displayed by selecting ‘Public GPS Traces’ via the layers icon at the right. There is a tab at the top of the osm page to upload your gpx.
You can use the josm editor to select ways and save as gpx. I have not got access now but you may have to create a new layer first and merge the selected ways in to the new layer first.
There is a web site called ‘waymarked trails’ that displays hiking trails that can be downloaded as gpx.
Among the numerous sites that you can use to draw gpx tracks, I find gpsies.com to be an excellent site to make gpx tracks for hiking as you can swap between osm, sat imagery and google and follow roads or offroad as you draw the track.
What I would really like to do is generate a route and save it as a gpx file that I can use in other software for example-FoxtrotGPS which uses OSM. By saving the gpx file in a folder, I can load it and the route will show in the FoxtrotGPS maps. Then the attached GPS will indicate my position
Then you use GPSies.com. Draw your route for walking, cyclling or car. Then export/download it as a GPX Track. It uses Graphhopper (which uses OSM) for routing.