Both of the datasets that you are using rely on special projection systems used by the MODIS system. While it is possible to use ArcGIS etc to read the data, it will often cause projection issues at various stages. Instead, you should convert the data to GeoTIFF and project it into a more suitable projection system, such as GCS WGS84 using the MODIS Reprojection Tool - the MRT can be downloaded here.
It is reasonably simple to use this tool, and also pretty straight forward to implement a batch solution that can help you process all 800 images in one go.
You can also use a polar coordinate system for better results in arctic regions.
In retracing your steps, it seems as if there is a bug in the input datalayers, resulting in MRT being unable to read them.
As for your work flows, going through 'Define Projection' into 'GCS WGS84' followed by 'Project Raster' into 'LAEA' the results come out as expected. The warning that ArcGIS provides makes sense, given how ill suited the LAEA projection is for global data, and thus, if you were to reproject the dataset back into lat/lon, you would get significant distortions. However, if you keep the data in LAEA, it should be okay. However, using global data that originates in a 0.05 degree climate modelling grid in a arctic application is not a good approach, due to how ill suited 'degrees-as-pixel-size' is in the arctic. Instead, you should rely on swath data and calculate derived layers yourself.