Well I read [1] [2] that they do have gravity... which is probably a more direct answer to what you're asking. I'm not sure what counts as "mass" for a photon per se, except that I assume there's an effective mass via the energy.
not exactly. As photons are constantly moving at the speed of light (citing the theory of relativity I don't really understand) they have no resting mass. But they have an impuls and create a force of impact (used in sun sails) from which a theoretical notion of ... mass (kinetic, perhaps?) can be derived. I'd like to know how it's derived originally, too though.
edit: if E=mc² with E(photon)=f•h and f=c/λ, then m=h/(λc) in vacuum. ... I hope that's correct. What's interesting, [m]=J/[a]/m
normally, people don't discuss mass. rather they are simply heavy. most don't even really care why. and those who do, conflate the idea with weight often enough.