Is the gradient at a layer (of a feed-forward neural network) independent of the activations of the previous layers?
Yes, as per @recessive answer they are indeed independent of the previous layers.
The goal of back-propagation is to trace the loss(error between target and network output) to specific weights in the network, and then tweak them to minimize this loss.
For this to be possible, the activation must be independent of previous activation functions(going forward in the network).
It is very helpful to have a good understanding of the back-propagation process when reading these papers, and I personally suggest watching this video to get a good understanding of it: