I had an edit block in the previous comment, but I kept getting it wrong, so here it is (hopefully correctly):

Let me add some explanation of why the size of the initialization weights is important. Let's imagine your inputs to a particular layer are uniformly distributed random numbers between 0 and 1. If you multiply them by random weights between -1 and 1 and add N of them together, the size of the result will typically be about sqrt(N)/3 (that's the standard deviation of the resulting distribution). For large N that means that your sigmoid function will saturate very close to 0 or 1 with high probability, and it's very hard to learn anything from there, because the sigmoid function is so flat in those regions that the gradients are pretty much 0. By using weights between -1/sqrt(N) and +1/sqrt(N), the typical linear combination will have zero mean and standard deviation 1/3, so the values of the sigmoid function will have a very small probability of saturating and you can still learn how to adjust the weights usefully.

The assumption that the inputs into the layer be uniformly distributed is not very important. If instead they are independent binary inputs with probability 1/2 of being 1 and probability 1/2 of being 0, the standard deviation of the random linear combinations with weights uniformly distributed between -1/sqrt(N) and +1/sqrt(N) is 1/sqrt(6) ~= 0.40825. So the situation is qualitatively the same.

I think it would be more useful to see the weights before and after one update, together with the computations that led to that update.

A neural network with equal weights in each layer will have equal weights in each layer after a step of gradient descent, so it will be as effective as a network with a single hidden unit per layer. This is well known.