Use "parts" in a master thesis report
So when is it appropriate to use \part? Any specific document length in pages or number of chapters?


It is appropriate when you feel it appropriate! Seriously though, assuming that there isn't any guideline from your university (otherwise, you probably wouldn't be asking the question), it's really a personal choice, the same than about the splitting chapter/section/subsection/subsubsection/paragraph. The structure of a document is there to help the reader, in particular for long documents. A part could be something that can be read separately (i.e., with no references to the other parts), or could be something containing some huge chapters; tt could reflect a chronological order (before, during, after), or a conceptual separation (theoretical, applied), etc. 

In case of doubt, talk with your advisor!


  When you use \part then a page is cleared after the text. Is this a style that is used in American PhD thesis documents? Why is this done?


As @Daniel pointed out, this is a rather technical question, that is more likely to be answered on tex.SE.