I’d like to know if any constelation of 4 charges is a quadrupole. I have a task where there are 2 positive and 2 negative charged particles in the corners of a rectangle. And this is a quadrupole. Is it still one if I place them differently or if I have 3 positive and 1 negative? About the second question: I’d guess that it’s not a quadrupole anymore but a more complicated dipole.
Yes, at some point in space their fields will sum to zero, for four charges this can happen in two locations. For example, if you have the charges evenly spaced in a plane ordered as (+,-,+,-) then you will have two quadrupoles midway between the charges at heights above the plane $\pm z_0$. If you have them ordered as (+,+,-,-) then the quadrupoles will be in the plane between the (+,+) section and (-,-) section.
It gets hard to imagine the field in your other scenario, I can imagine one scenario where the positive charges are on one side and the negative on the other. This configuration will give you two quadrupoles and a dipole. I'm not sure if this is the case generally.