Event "On Spawn"?

Is there an event "On Spawn" called by the object whe it gets initialized?

Not really.

In fact, if you want to act on objects that you already placed in your layout in edittime, you do it through through "Start of layout", making subevents for your picking needs.

Else, when you are spawning your object in your code, that is the moment to initialize it. (right after the "create object" action, if you apply code to the object type that just was spawned, only this very object is picked).

Sometimes I indeed need to apply stuff to a newly created sprite.

Just today I did something like that :

Add a boolean onCreate with default value true

Mouse: On Left Click
    -> Create Sprite on layer 0 at 0,0
Whatever condition you need to check
Sprite: Is onCreate 
    -> Sprite: Do stuff
    -> Sprite: set onCreate to false

Thanks Guys! That helped to solve the task!