On each side of this dormer there is a small gutter which drips onto the roof and into the longer gutter and down the side of the house. If this short gutter wasn't there, wouldn't the water go to the same place anyway? Follow up, if the short gutter is a good idea, should I connect it to the longer gutter or leave it as is?
Since the dormer extends out past the main roof, some (looks like about half) of the water would fall to the ground below. This could cause problems with drainage, puddling, erosion, and possibly lead to water infiltration.
The gutter should be there, but there should be a downspout running from the outlet to the lower gutter (as described in this answer). As it is now, you will likely see deterioration of the shingles, where the water flows out of the outlet and across the roof. The rate of deterioration will depend on the volume of rainfall in the area, among other factors.