Frankly whether a node is processed depends entirely on your templates and the built-in templates (which can be overridden). If a template for a foo element nodes does then the child nodes are processed, if it does then its ancestor elements are processed, just to give two examples. Of course if you write a template for a certain element that does not use apply-templates for further processing then processing stops.
None of these statements are true. Different templates may be selected for execution on the same node -- this all depends on the instructions that result in selecting a particular template for execution. There is a great flexibility in how templates can be applied and selected, such as import precedence, priority, modes, ..., etc.