The distance from a point $(x,y)$ to the vertical line $x=-2$ is the horizontal distance $|x-(-2)|=|x+2|$. The distance from $(x,y)$ to the vertical line $x=6$ is $|x-6|$. The problem says these two distances differ by 4 units, so their difference could be either $4$ or $-4$. To represent both possibilities in one equation, take the absolute value of the difference: $\left||x+2|-|x-6|\right|=4$. Therefore, choice A is equivalent to the described graph.

Because $x>y$, the expression $|x-y|$ becomes $x-y$. So the system is

Add the equations to eliminate $y$:

so

For an equation of the form $y=|x-h|+k$, the graph is a V-shape with vertex at $(h,k)$. In the table, the smallest value of $y$ is $4$, and it occurs when $x=3$. Also, the values are symmetric: $y=6$ at $x=1$ and $x=5$, and $y=8$ at $x=-1$ and $x=7$. That means the axis of symmetry is $x=3$, so the vertex is at $(3,4)$. Therefore, $h=3$.

Distance on a number line is represented with absolute value. The drone's distance from the base station at $0$ is $|x|$, and its distance from the weather sensor at $18$ is $|x-18|$. The rule says the distance from the base station is exactly $6$ kilometers less than twice the distance from the weather sensor. Twice the sensor distance is $2|x-18|$, and $6$ less than that is $2|x-18|-6$. So the correct modeling equation is

Because point $P$ is on the line $x=4$, it has coordinates $(4,k)$. The distance from $(4,k)$ to $(4,-2)$ is entirely vertical, so it is the absolute difference of the $y$-coordinates: $|k-(-2)|=|k+2|$. Since that distance is $7$, the equation is $|k+2|=7$. Therefore, the correct answer is $A$.