One player must be named as goal keeper. That individual must wear a shirt to distinguish him from his team mates and the opponents.
The goal keeper, however, may play anywhere on the field they wish.
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They wouldn't want to, If a goalkeeper gets sent off or goes off injured they will instantly replace them.
Yes teams can play without a goalie. It is usually for short amount of times. I.E. not an entire game.
Yes. A team can play with a goalkeeper, a field player with goalkeeping privileges, or a field player without any special privileges.
One player on the field for each team must be named the goal keeper. This player must wear a uniform to distinguish them from both team colors and the match officials.