About viewController's "viewDidLoad" and "viewWillAppear" methods

Question 1

I would like to add to Caleb's answer: Don't confuse the view controller and the view! The name viewDidLoad clearly indicates that the method is invoked after the view has been loaded. It is the view controller that does the loading.

Some pointers regarding the lifecycle of views and the order in which messages are sent:

Not an official Apple document, but I find this diagram really useful because it includes pretty much all of UIViewController's lifecycle overrides.
In the section Resource Management in View Controllers from Apple's "View Controller Programming Guide" there is a flowchart that depicts how views are initially loaded. It explains loadView and viewDidLoad, also in conjunction with storyboards.
The section Responding to Display-Related Notifications from Apple's "View Controller Programming Guide" explains how to respond to views appearing and disappearing (viewWillAppear: et al)
If you are planning on implementing a container view controller: The UIViewController class reference has a good overview of how messages need to be sent by your subclass.

I'm stopping here. You can find more stuff yourself by googling for "uiviewcontroller life cycle".

Question 2

-viewDidLoad is called when the controller loads its view, which is not necessarily right after initialization. View controllers don't load their views until they need them, either to display or for any other reason.

-viewWillAppear is called just before the view is displayed. This will be after -viewDidLoad, but you don't know exactly how long after. -viewWillAppear is called every time the view is displayed; -viewDidLoad will only be called a second time if the view is unloaded at some point (such as didReceiveMemoryWarning). These days that's unusual, but it can happen.

Or if the viewController is set to nil, which can usually happen if a view controller is kicked off the navigation stack, and therefore next time it is brought to the navigation stack it needs to call -viewDidLoad again.

Question 3

I thought that, firstly, viewDidLoad is called when the viewController is loaded for first time (as the name indicates), and inmediately after the init method

No. The name indicates that the controller's view has been loaded (not the controller itself). Actually the docs state that this method will get called after the view hierarchy has been loaded into memory (either via loadView or through a nib for example).

Then, I thought that once viewDidLoad returns, viewWillAppear is called

Again, no. loadView (and as a consequence viewDidLoad) method will get called the first time that view property is to be accessed and is nil (which is the case when you're initializing a controller). Think of this simple scenario:

MyViewController *vc = [[MyViewController alloc] init];
UIView *view = vc.view; // <= loadView & viewDidLoad will fire but it certainly didn't appear...

However, while developing I make the impression that there is no order when calling viewDidLoad and viewWillAppear...

Well there is an order. We know for sure that viewWillAppear will always be called after viewDidLoad (if both of them are to be called of course).

Question 4

As you said, ViewDidLoad is only calling once after loading the view. So we can initialize the instances in the viewDidLoad. It is mainly meant for the initialization.

viewWillAppear will invoke whenever we reach to this view. So if there is any changes in UI, we can done it in viewWillAppear.

Question 5

I ran a trace on when all these calls are made: http://thecodist.com/article/ios_arc_storyboards_and_uiviewcontroller_trace