I know that there are many many posts about this type of topic already, but all solutions did not work out for me so far.
I have a shapefile with polygons, which is described here and can be downloaded here (there is also a paper published in the Journal of Urban Economics, see this link). It comes as a geopackage. As the first link shows, the projection is World Mollweide (EPSG: 54009). I want to calculate zonal statistics for these polygons with various raster files, e.g. air quality (downloadable here (as .nc) or here (as .asc), which use the WGS84 projection. However, I just cannot manage to align the raster files with the shapefile.
I tried to save the polygons as a shapefile, assigning to it the WGS84 (EPSG: 4326) CRS. However, using QGIS 3.8 this results in a shapefile without any polygons/observations. With ArcGIS Map 10.7 or ArcGIS Pro 2.5, it tells me that I cannot export the shapefile. While ArcGIS aligns the CRS on the fly, it does not allow to compute zonal statistics (because of some mysterious error 99999).
Any suggestions how to solve this problem?