How long should I saute garlic

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.

Yes you need to devein them. The 'vein' is in fact no such thing; it is the intestine, and is thus filled with waste matter.

Yes, you need to peel them. Shrimp shells are indigestible, and serving shrimp (unless meant to be eaten with the fingers) with the shells on is, to my mind, indefensible. There is simply no good way to eat unpeeled shrimp with cutlery.

I would personally eat them cold, as I see no point in re-cooking something that has already been cooked. This will make the flesh tougher.