Let the starting point be the origin $(0,0)$.

1. The person faces East and travels $4$ km straight. Current position: $(4, 0)$. Facing: East.
2. The person turns right. Since they were facing East, a right turn means they are now facing South. They travel $3$ km. Current position: $(4, -3)$. Facing: South.
3. The person turns left. Since they were facing South, a left turn means they are now facing East. They travel $2$ km. Current position: $(4 + 2, -3) = (6, -3)$. Facing: East.
4. The person turns left again. Since they were facing East, a left turn means they are now facing North. They travel $3$ km. Current position: $(6, -3 + 3) = (6, 0)$. Facing: North.

The final point is $(6, 0)$ and the initial point is $(0, 0)$. The minimum distance between the final and initial point is the straight-line distance: $Distance = \sqrt{(6 - 0)^2 + (0 - 0)^2} = \sqrt{36} = 6 \text{ km}$

At the final point, the person is facing North.

Therefore, the minimum distance is $6$ km and the direction the person is facing is North. This matches Option D.