Answer:
I am a band that does not perform sing or act.
What is shaped like a box, has no feet, and runs up and down?
Against the sun I protect your eyes and am often worn on the face of spies
Squeeze me and I cry tears as red as my flesh, but my heart is made of stone. What am I?
I am whole but incomplete. I have no eyes yet I see. You can see and see right through me. My largest part is one fourth of what I once was.
Those with eyes bigger than their stomach will definitely leave the restaurant with one of them. It is named after a house pet as well.