Hi Konstantin,
I looked the data you provided. Out of the 8 files, only one h5 file has variables of Geo2D type, all other variables in these files are either 1D or 2D (which does not have any coordinate defined). Mosaic dataset or raster layer does not support 1D array. ArcGIS can read a 2D array as raster, but as these 2D array does not have any Geo info, it will not make any sense to display in map. These are as designed.
ArcGIS supports reading HDF5 that has well defined coordinates, such as HDFEOS and a few products that follows CF convention, looks like this product is not among those. The file (GMT_npp....h5) has a height variable of Geo2D type, but ArcGIS reads it off 180 degree. one workaround to display this Height variable is that you convert the Height variable to a TIFF (Add Multidimensional raster tool and Export to a tif raster, or use the Extract Raster To Subdataset GP tool), then use the Shift GP tool to shift the raster -180 degree in X direction. Hope this helps.