Justification for values used in backpropagation
The objective of back-propagation is to isolate the effect of each weight on the total error. Once the effect is isolated, each weight can be changed individually such the total error is minimized.

During back-propagation, the effects of other weights are automatically isolated. Take, for example, the function:

$$
E_{total}(x) = w_1x + w_2x^2 \\
$$

Then

$$
dE_{total}(x)/dw_1 = d(w_1x)/dw_1 + d(w_2x^2)/dw_1 \\
dE_{total}(x)/dw_1 =x + 0
$$

So, you see, back-propagation ignores the effects of other weights by definition of differentiation. When you back-propagate with respect to $w_1$, all other $w$s are considered constant.

$dE_{total}/dw_i$ is the change in error as you change a particular weight $w_i$ and only that weight. So it makes sense that you would want to change $w_i$ in the opposite direction of that change i.e. $-dE_{total}/dw_i$.