Removal of everything except mesh, or extraction of mesh from Blender file to a new one I want to have just the mesh part of the character I downloaded from here. I don't want any of the UV-mapping, materials, rigging, and stuff. 
I'll add that stuff later with my own materials. In any case, I can't seem to delete everything. The old stuff is messing with the new stuff; it's like I have to go really deep to remove everything.
Open your downloaded file, create a cube, name it "onlyMesh", go to the mesh tab in the properties panel, switch the cube mesh to the character mesh, delete the whole original character, save, close, reopen your file.