In QLever, click on “Map view” to open the results in Petrimap, then click on the “Export as GeoJSON” button to export a GeoJSON file. Overpass turbo’s GPX export option uses the togpx library to convert the data from GeoJSON to GPX. This library is also available as a command line tool, or you can use any online converter or desktop GIS tool such as QGIS.
This may not be as convenient as the one-click GPX option in Overpass turbo, but someone could trivially build a webpage that ties it all together.
Yes, if you select a location on the map, OsmAnd links to Wikipedia and embeds Wikimedia Commons images, based on Wikidata if the image isn’t explicitly tagged on the element. Organic Maps also links to some travel websites, though not without controversy.