In the first screenshot, it seems that the penguin has no material, unlike in the last screenshot where the penguin has 3 materials assigned (and there it shows the texture)..?
Could it be just this?
Following you request in comments/chat, I did a test:
1.I downloaded your file, saved on my desktop
I created a new default file in blender
I appended just the penguin mesh from your file (I found it's named "untitled" in your file, under "mesh" folder.
Then I selected my default cube object, and switched to the "object data" in its properties panel
From the "browse mesh data to be linked" selection tool, I selected the previously imported penguin mesh, named "untitled".
Now my "cube" object has a penguin shape. Your UV grid also gets imported.
Setting viewport shading to "textured" shows the textured penguin!
So, I think the object in your file was corrupted, but the mesh is fine.
You can correct your file in this way, I think.
Btw, your mesh has arms normals inverted, select all mesh vertices, and fix them (CTRLN).
This is my file: