Gravitons are at best hypothetical particles. Their existence is presumed from the other (known) elementary particles that "mediate" the other forces. Nobody has seen one and it's probable nobody will because of the huge energies likely required to make one appear. It's not clear to me how "mediating" relates to interacting. I don't think it's a matter of swarms of gravitons hooning about doing the work of gravity on any mass they come in contact with. We run slap bang into the usual wave/particle duality problem.
Re having mass, AFAIK the mass that may be associated with a graviton is extremely small.
Similar to a neutrino. Originally these beasts were supposed to have no mass, but it turns out they probably do have a very small mass, which allows them to flip between the three different types of neutrino. This explains why the neutrinos apparently coming from the core of the Sun were not the type expected from the nuclear physics going on there. It appears they've transformed themselves into another type by the time they get to Earth. See for example,
http://www.ps.uci.edu/~superk/oscillation.html
DN