I'm not a traffic engineer. Help me understand this intersection that I (frustratedly) drive several times a day. 

When traffic on the east-west road has red lights, the traffic coming from the north is turning either left or right (shown with the green arrows). 

On the east-west road, there are 2 lanes of cars facing west and waiting for the red light. The left-hand lane is straight only. Of the cars in the right-hand lane, about 40-60% eventually make a right-hand turn (shown with the dotted blue arrow), depending on the time of day.

When the east-west traffic has the red, is there some reason there can't be a green arrow for the cars in the right-hand lane to make a right without having to stop first (as you would for a traditional right-on-red)? Obviously if the car at the front of the line is planning on going straight, it's irrelevant. But at certain times of day, most cars are turning, and it's not uncommon to have the first 8 or 10 cars all have their turn signal on. More cars could make it thru the intersection if they could confidently cruise on thru rather than each car pausing to check surroundings.

There are no pedestrians crossing then, because of the traffic coming from the north. Is it against code to have a green turn arrow in a lane that isn't a dedicated "turn only" lane? Or am I correct that the city is missing an opportunity to make the intersection more efficient?

Why no green arrow for cars turning north?