Should all subsections of a paper start on a new page?
Depending on the length and complexity of the work, you choose the top level sectioning.


For short works -- scientific paper, lecture notes &lt; 80 pages, etc., you in general use only sections and lower.
For longer works -- thesis, long lecture notes, books, etc., you in general use chapters, sections and lower.
If your long work has multiple sort-of separated parts, you can add parts -- notice, however, that usually, part numbering is independent. Therefore for instance Part I contains chapters 1, 2, 3 with some sections, and Part II contains chapers starting from 4, ...


Now, the standard way to go is to start parts and chapters on a new page, and leave sections running through the pages.

Needed to say, most standard publications (articles, conference abstracts, theses, books by standard publishers, ...) have to comply with some guidelines provided by the publisher, and these guidelines are usually very restrictive.