How does the production of pure, crystalline silicon via the Czochralsky or Siemens process? I have a few questions regarding the production of pure, crystalline silicon:

As far as I know Silicon usually appears in nature in the form of sand or $\ce{SiO2}$ and then is split apart and brought into the form $ \ce{Si + 2CO} $ ; 
I am imagining now a hot glowing molten mass. 
However then I don't understand the following methods to separate the Si from the elements you don't want to have, like iron, carbon etc.
How do you remove the impurities, why do those methods work?
For example you use a few other chemical reaction to bring Si into several other forms like $ \ce{HSiCl_3} $ and split them apart later on. 
I don't understand at the moment, how this helps to separate those materials as in: There is still this big mass with lots of different chemical connection you don't want to have. 

I also lack an actual understanding of the Czochralski process to further purify your mass of silicon. I guess want has to accept that silicon likes to join pure crystalline silicon and adapt to its structure. However I don't see, why one is not using this method beforehand for example, if one already has raw silicon you get from splitting up $ \ce{SiO_2} $ if it works so greatly. And if it not does work so greatly, why don't you have other impurities ?
How does trichlorosilane $\ce{HSiCl3}$ help in the purification of silicon?

Trichlorosilane is a highly volatile compound with a boiling point of 32 °C. This means that it can be destilled off the raw material and separated from impurities.

The purified trichlorosilane can either be decomposed thermally

$$\ce{4 HSiCl3 -&gt;[\Delta] Si + 3 SiCl4 + 2 H2}$$ or reduced with hydrogen

$$\ce{HSiCl3 + H2-&gt; Si + 3 HCl}$$

This does give you rather pure silicon, but it is, at best, polycrystalline.

What's the Czochralski process good for?

Note that this in't just a melting/cooling cycle! 

The crucial point in the method is the use of a rotating, monocrystalline seeding crystal, dipped into (previously purified) molten silicon. The goal is to obtain huge, monocrystalline silicon rods of defined diameter, that can be used in the production of wafers.