Yes you should format a new memory card you brought. U should not format only in the computer and also you should format it in the mobile which you are going to use the card. if did so only the default folders for the mobile will appear.
It may work without formatting, but it is best to format a new memory card. Because the company which u provide the memory card may have some type of softwear. Its good to remove the unwanted matters in your card by formatting it.
there is no need to format your new memory card. directly use the memory card , if any virus or trojans affected your memory card then format your new memory card.
Mostly you don' need to. Both if you do, you will understand it because your device will not recognize the card. All devices such as cameras have format fuction in their menu.
That really depends on the device your using it in. Some require the card to be formatted before use, others do not. It seems as if the most recent ones I've purchased have not required formatting. I just popped it in and went!