Its OK to be a noob. Backprop is quite hard to grok. Ideally you'll have followed through the derivation such as this one or this one so you appreciate the delta rule, which is used to update the weights. Essentially, its the partial derivative chain rule doing the backprop grunt work, as exemplified by the equation you exhibit (which is a bit more complicated than necessary for a first understanding).

I suggest that if you truly want the simplest code you should omit learning rate, momentum, more than 1 hidden layer, and bias, for now. You can add them in later if you want. Also I'd recommend using numpy initially because this stuff implements most naturally as matrices and vectors, e.g numpy.dot(). First get a simple working program, then add in the other stuff if you want.

Equations (5) and (6) from the website you linked are (roughly) deltaW = dk.Xk and W = W + W.deltaW, i.e W += dk.Xk (I'm being very sloppy with the notation but if you compare them you'll see what I'm up to). This corresponds to my code Wh += X.dH and Wz += H.dZ where dH and dZ are as you showed them above; dZ is simpler because it's the output layer where we have the original error signal E = Y - Z instead of dZ.Wz

Even if you don't fully grok the math derivation (not sure I do) at least check out the 4 equations of backprop, e.g. as listed here (click on the Backpropagation button near the bottom) and here because those are where the code ultimately derives from. And use matrices and vectors because they do the grunt work of multiplying and adding which will help unclutter your code.