Should one always use the whole garlic clove?
The green in the center can have a bitter taste, and many chefs prefer to remove them. If the taste and color don't bother you, it will not cause problems for the dish. 

The brown end tends to a have a harder consistency than the rest of the clove, so may cause a problem depending on how you cook the garlic. If you're cooking it in a manner that leads to a very soft and mushy piece of garlic, you'd be better off discarding it. (Boiling in soup, for example.) It probably would not make much of a difference if you were browning the garlic for a topping.