Does entity_save() trigger the update event? Not quite sure if calling entity_save() from code will trigger the update event or not. 

So, does it?
Assuming you're asking whether or not entity_save() invokes hook_entity_update(): no, it doesn't.

The controller class (or save callback function) for an entity type may well invoke it (for example node_save() does), but you can't rely on it for an entity type without checking the code first.