Feeling overwhelmed with this math explanation, what videos, courses, and books should I learn to help me digest this?

Question

I'm recently trying to learn about deep learning, and stumbled across this excellent video explanation by Hugo Larochelle

Neural networks [5.2] : Restricted Boltzmann machine - inference

But once I got to the mathematical explanation on probability of hidden units given inputs ( $p(h\mid x)$ ), the mathematics there, I'm just not sure how and why some of the transformation even worked. I paused and restarted the videos multiple times, but still there are a few questions unanswered:

1. I understand that the first line there was from Bayes theorem, but I don't get how the sum of joint probability of inputs and all possible hidden layer vector (denoted by $p(x,h-prime)$ could replace $p(x)$ that should be the denominator there (he mentioned it briefly in 5:01).
2. In 8:00 - "...what this means is, if I'm summing over x-capital-H all the terms here in this product are constant with respect of h-prime-capital-H, except for the last one, except for h-prime-h, so the last term, for j equals capital-H. So it means that all of the other factors in this product I can actually put them in front of the sum and just perform the sum over the last hidden unit of the corresponding factor in this whole product here. And once I compute this, this is a constant with respect to all of the hidden units so I can actually put it in front of this whole sum, and in this way I could actually write down this nested sum here as just a product of the sum over the first hidden unit times the sum over the second hidden unit and so on..." That is a load of information there for me. I don't understand what that explanation actually mean, can anybody help me rephrase that please? And same question as above, what math courses should I learn first to "get" this?
3. Counting from the top, I don't get how he got from 5th to 6th line there. What are the rules of products and sums that allow these sums to turn into a product?
4. 10:56 - How did that end up become a distribution?

Sorry that it's a lot of details, but I would really appreciate any help in this, I'm very stumped.