I really liked the question. Yes, we sum over derivatives. First of all think what backpropagation is trying to do: finding the affect of each parameter on the loss.
So as you said:
the same filter is used multiple times on the input while convolving
meaning that each kernel affects the final loss in several ways, so those affects should be summed together,...
The claim that Neural Network with a single hidden layer can model any functions is proven in Cybenko's Approximation by superpositions of a sigmoidal function.
check also: https://en.wikipedia.org/wiki/Universal_approximation_theorem
The thing is that the neural network using sigmoidal functions, which ...
Don't know if you have this doubt anymore, but this would be helpful for those who are facing similar problems-
You will need to find the correct weights with which you add these two loses by hyperparameter search. That is, find the best $\lambda$ for the loss-
L = Loss_1 + \lambda(Loss_2)
Here $Loss_1$ and $Loss_2$ can be any losses. Here, we take ...