Normally the per-vertex normals are stored in the GR2 file, and when you export/import the model to/from DAE it'll automatically import the normals from the DAE. If normals were not exported to the DAE, they will be recalculated. (Please note that these automatically calculated normals are generally less precise than those specified manually and may contain errors, therefore it is recommended that you export the normals from whatever modeling program you're using.)
There is a normal map texture (called something like Chest_NM) which contains the per-texel normals encoded into the texture. If you didn't assign a normal map to the material after importing the mesh into the editor, it'll only use the per-vertex normals.
In addition, there is a physical map (Chest_PM ?) which contains the physical properties (anisotropy, metalness, roughness) of the material; this also needs to be assigned to make sure that the mesh will look fine ingame.
It seems that in your first two pics there was no normal map assigned at all, and the normals all pointed in one direction (hence the "2d" look and the grayness/blackness when looked at from specific angles). I'd say that in your third shot the physical map is missing, since the reflectance parameters seem to be the same on all parts of the mesh.
