Does a 3 d vector exist in three dimensions at once? It seems to me that a vector always changes axis along which it is to fit into the scenario. For example: Electric field in $x-y$ plane when passes through $y-z$ plane only uses it's $x$ component.
Yes, a vector exists in three dimensions at once. Imagine we apply force to a body diagonally then it will also move diagonally, which means it is moving in both x and y directions. If you block that object from moving in only y direction then you will see it is moving in only x direction while we are still applying the force diagonally. If we know the amount of force we are applying in this case then by resolving it, we can find that how much of the total force is acting in x axis and in Y axis because some part of the total force is acting in X direction and some part is acting in Y direction at once. Same goes for three dimensions.
Does a 3 d vector exist in three dimensions at once?
Absolutely, a general vector is $$\vec r=x \hat i+y\hat j+z\hat k$$ in Cartesian coordinates. This vector has a component in each of the three dimensions and all components are mutually orthogonal. Since the vector $\vec r$ is equal to the sum of these 3 components then it must exist in three dimensions at once.