Is there any meaning of weight of the earth? Can you think of two particles
which do not exert gravitational force on each other?

Yes.

Consider two earths.In this case the weight of the earth is the force one earth would experience if one is on top of the other.
Answering your second question, Gravitaioon is a universal Law. It means it applies to every body with a mass. Thus, consider a photon at rest. In this state it has zero mass. Hence it does not experience any force, nor does it apply any force on other particles.