First you need to understand what a prototype is.
In Javascript, almost everything is an object.
Edit (thanks @Mathletics):
Primitive booleans, numbers, and strings are not objects
When you create an object in Javascript, by default, its prototype will be an Object.
So, if I do:
var obj = {};
console.log(obj.__proto__)
This yields:
Object {}
Pause:
To understand the difference between obj.prototype and obj.__proto__, please, refer to this thread.
The next step is to understand the concept of prototype chaining.
This is how inheritance is structured in Javascript. Basically, a prototype is the "parent" of another object, from which it derives methods and properties.
Imagine a scheme where a Rabbit inherits from Mammal, which inherits from Animal. Representing this in Javascript prototype chain would be like:
(Rabbit) -> (Mammal) -> (Animal) -> (Object) -> null
With code:
function Animal(){}
function Mammal(){}
function Rabbit(){}
Mammal.prototype = new Animal();
Rabit.prototype = new Mammal();
If you are able to understand these concepts, you might find the answers to your questions yourself, but I'll try to make it clearer:
1) why one of the following is false and the other is true?
User.prototype.hasOwnProperty('name'); // false
User.hasOwnProperty('name'); // true
You should read the hasOwnProperty method reference:
Every object descended from Object inherits the hasOwnProperty method. This method can be used to determine whether an object has the specified property as a direct property of that object[...]
This is a way to assure that the property you are looking for is defined within the object itself, not deeper in the prototype chain.
A contrary example would be:
console.log(User.hasOwnProperty('constructor'));
Since the constructor property is defined deeper in the prototype chaing, it is not an "own property" of User object.
In your case, name is a property defined within your User object, but not defined in the context of the global Object.
2) what is de difference between the followings:
User.constructor;
User.prototype.constructor;
The fist like takes the constructor of User, the second takes the constructor of its prototype (in this case, the global Object).
3) what happens with the User.constructor and
  User.prototype.constructor if I override the prototype property like
  this:
User.prototype = {
    changeName: function(newName) {
        this.name = newName;
    }
};
Please, read this thread.
TL;DR: when you assing directly to the prototype, you are breaking the prototype chain an creating a new one. In the Rabbit example, if you at some point did:
Rabbit.prototype = {
   // ...
};
Your new prototype chain would be:
(Rabbit) -> (Object) -> null
So, in your first example, User.prototype.constructor would yield:
function Object{}
In your second example, it would yield:
function User(name)
Got it?
4) is User a function or a prototype or what?
User is an object whose prototype is a Function object, whose prototype is a Object object.
Drawing it:
(User) -> (Function) -> (Object) -> null
Hope I've been clear.