Can any magnet (or even one much stronger) of the Earth divert the cosmic rays as Earth's?
No, because even if the strength of the field may be higher, the size of magnetic field is too small. Although the force that your local fridge magnet exerts may be larger than the one of the Earth, it does so only for a much smaller area. Even if you brought your magnet outside of the magnetosphere, charged particles would pass straight by it and only have a tiny deflection, before the particles are outside of the fridge magnet's reach of influence again. The deflection would be the integral of force over path length of the particle inside the field.
Do these magnets somehow "interfere" with Earth's?
In the immediate surrounding of the fridge magnet, the fridge magnet magnetic field is stronger than the one of the Earth, but at any larger distance, the Earth magnetic field is stronger. The larger Earth magnetic field is not significantly affected.
Maybe because the size (not the power)?
Yes, size does matter.
In the magnetosphere, the only significant magnetic field is that of the Earth. The field of your fridge magnet is non-existent there, as its range is very small. Charged particles are diverted by the magnetic field in the magnetosphere. When they do enter the Earth atmosphere, many of the remaining one are absorbed by the atmosphere (forming the northern lights).
