Why is it incorrect to to add accelerations when something is accelerating downwards and gravity is downwards? I know it is related to apparent weight and how one feels lighter when accelerating downwards in an elevator. But I have no idea to explain it.
 A: We'll define up as positive. So if a numerical value is supposed to be upwards/downwards, and we actually drew the arrow as upwards/downwards it will be positive. If the actual direction of the numerical value is the opposite of the direction we have made our arrow it will be negative. Normally, we would define our arrows for all known quantities in the direction they are in so we can use positive numbers. It's just simplest that way, but if we chose to, we could draw the arrow in reverse and use a negative value instead.
Where this really matters is for unknown values where we do not know the direction until we solve it. In this case, we have to draw an arrow and assume a direction. If it turns out to be positive, we assumed right. If it turns out to be negative, we assumed wrong and it is actually in the opposite direction of the one we assumed.

$ma = \Sigma F $ (notice how the diagram above mirrors this equation)
$-ma = N - mg$
Notice how the LHS term is negative? Because we defined negative as down.
$N = m(g-a)$
So you can see here that the left side of the diagram is the left side of the equation, and the right side of the diagram is the right side of the equation.

*

*$a$ is the acceleration of the object (which is the same as the
acceleration of the elevator)

*$mg$ is the force that gravity exerts on the object, NOT the weight of the object

*$N$ is the normal force, the reactive supporting force that the floor of the elevator exerts on the object, in other words the weight of the object.

