Question: Let's imagine a massive object $M$, about to enter in collision with Earth on west Europe. This object is way more massive than Earth. Before the collision, is there any objects on Earth that will be enough attracted to $M$ to move in the direction of it before the impact ?

Relatively light object would "levitate" from Earth point of view before the impact. Example of relatively light objects :

* Gas, particles and liquids: atmosphere, water, sand.
* Organics: animals, plants.

Of course this is true for Europe, would be interesting to know what happens on the other side of the planet. Would buildings crush?

I suppose the velocity of a given object changes when the gravity's force of $M$ on it is bigger than the gravity's force of Earth on the same object.

Answer: 
If the object was much much wider than the earth, you would accelerate toward the object at the same rate. This is the same idea as you and a pebble accelerating toward earth. If you have ever done this experiment, you know your accelerations are equal.

$$
F = -G M m/r^2 = m(-GM/r^2) = ma
$$
$$
a = -GM/r^2
$$

Thus acceleration is independent of your mass or the earths mass. It only depends on the objects mass and separation. The distance between you and the center of the earth would be the only thing that could cause a discrepancy between your accelerations. This effect could be major or minor depending on how wide the object is in relation to the earth. If it's much wider than the earth, the difference will be negligible. If it is comparable or smaller, the difference will be pronounced.
You're basically talking about the Earth going inside the Roche limit of the other body: the near side falls in faster than the rest of the planet (it's lifted into the sky), while the far side is flung off the planet (into the sky). Where the other body is on the horizon, stuff is squished into the surface. This is basic tidal deformation, and it looks a lot like The Rapture--at least dynamically.