Reason behind the sum of rate factors for calculating cost function derivative
First, disclaimer: you're presenting fairly hairy computations and it's a little hard to read your question. I'll try answering nonetheless.

i don't understand exactly how can we calculate that slope of the cost with respect to that weight by summing [...]

If I interpreted the sums correctly then the sums in your question seem to deal with the Jacobian matrix that is used when applying the chain rule in multiple dimensions.
Intuitively, the weight $w_{j,k}^{l}$ will be referenced by all the paths originating from the neuron utilizing $w_{j,k}^l$, and will then be expressed by nodes in the &quot;final&quot; expression of $C$ via multiple linear combinations (sums). I'm fairly certain that if you actually followed the gradient $\partial C\over\partial w_{j,k}^l$ starting at the sum $C=a^L_{1}w_{2,k}^L+a^L_{2}w_{2,k}^L$ it would work out to be the sum you show. This also, I believe, would apply to the second sum you present.