Should I install a gutter above my garage door?
Gutters are for water management. If you have a paved drive and it drains well then you don't need gutters in front of the door. If water tends to flow back toward the garage when it runs off the roof then you should put a gutter in. Having a gutter will mean less splash on the bottom of your garage door which may help the door last longer. 

What others do with their gutters is fairly irrelevant to your situation.