Definitely.
Humans don't digest all plant material very well. If you can get nutrition out of it, it's more efficient if you can eat the plant material directly. If you have to feed it to a cow, and then eat the cow, a lot of the nutrition will be lost in the transition. But if you can't get nutrition out of it, and the cow can, then feeding it to a cow would be sensible.
No.
There's no such thing as an "organic cow." A cow is a cow, regardless. She will eat what "normal" cows eat, which is defined in the related question posted below.
No, a cow will not eat meat, although there will be some curious cow that would probably try it, cows do not and will not eat meat.
No, never. Cow paddies are cow feces.
You don't, it's disgusting if you eat a cow pat!
Only if the cow is dead. Vultures don't usually swoop down upon an alive cow and eat it.
No, a cow eats grass.
The cow.
cow balls cow balls
Humans, and any other carnivore (or omnivore) that is higher up on the food chain than a cow is will eat a cow, either by killing and eating it, or scavenging a cow's carcass.
If you eat a sacred cow, you will be subjected to the disapproval of those who considered the cow to be sacred.