What are the activation functions in Convolutional Layers for?
1 - Activation functions are non-linear functions. These are added in between layers which are simply Linear transformations.
Example without activation function:
ConvLayer1(Input) -&gt; ConvMaps1 
ConvLayer2(ConvMaps2) -&gt; ConvMaps2

Mathematically, this would be $I_{nput} \circledast K_{ernel_1} \circledast K_{ernel_2} $,
which is equivalent to $I_{nput} \circledast K_{ernel_3} $,
where $K_{ernel_3} = K_{ernel_1} \circledast K_{ernel_2} $
That means that creating two layers would simply waste computational effort and give no gain whatsoever.
2 - Yes, if you keep the same image Width and Heigh and just stack more channels, there convolution maps will grow out of control, this is why you compensate this with Pooling Operations and use of large strides. Also, increasing the number of parameters without proper care and architecture design usually leads to overfitting.
3 - Backpropagation is simply a computationally efficient and convenient way of expressing the Partial Derivatives of the function and acts exactly as it would if you just took the derivative of the whole function at once. There is plenty of content online on it, you probably didn't understand the process of weight updating.