Store in an open container at room temperature, in the shade. Whole heads last longer than separated cloves.
Read about Botulism. It's a deadly disease you can get by improperly storing garlic/oil mixtures or other things.
Garlic with access to sunlight will start sprouting. A little bit of green won't hurt you but the flavor is different.
Some cloves might get tiny brown spots. Cut them off and use the rest of the clove. If it's more rotten throw it away.
The outer peel of the garlic clove should be removed. But the peels of the individual cloves are fine, if you are only using the garlic for flavor and removing it, or if you're baking the whole head.
To peel, first separate into cloves, then cut off the brown part of the clove and the peel should come out easily. You can also smash it instead of cutting, but cutting allows you to look inside and see if it was close to sprouting.
Garlic can be used in various shapes:
The factors that inform your choice are: