All the bottom part of the mesh where it is highlighted on the UV map, has to come off where it is overlapping the fur. Unless textures on things are exactly the same, overlapping UV's cause all sorts of visual issues.
I checked the mesh, it looks fine for me here. The reason why there are pieces of flooring on your texture is because the person that made the mesh used a picture as a texture for their mesh (since they based their mesh on that picture), which was a dog lying on the floor, and when they cut out the dog from the picture to put use as a texture on their mesh and they didn't cut the floor out, but the mesh seems to be fine.
Meshes will look messed up sometimes when you decimate them to lower the polygon count, which is what I see they've done with this mesh. As far as the weird shadows that you mention, they seem to be weird shadows due to the actual shadows from the dog laying on the floor in the picture that they used as texture, unfortunately, you can't get rid of those unless you re-texture it.
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