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

Add the equations to eliminate $y$:

so

The rule for the device is $d=|p-12|$. The table shows that when $p=8$ and when $p=16$, the reported distance is $4$, so the fixed distance being described is $d=4$. Substituting $d=4$ into the rule gives

This equation represents all positions that are $4$ units from $12$. Solving confirms that:

so

These are exactly the positions shown in the table, so this is the correct equation.

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 $A$.

Being 7.5^4\mathrm{C} from the target temperature of 12^4\mathrm{C} means the reading satisfies

This gives two possible values:

So

Because the reading was below 10^4\mathrm{C}, the only value that fits is $4.5$.

Substitute the point $(1,6)$ into $y=|x-4|+k$:

So $k=3$. The equation is then

The $y$-intercept occurs when $x=0$, so

Therefore, the $y$-intercept is $7$.