Photons have no rest mass, but one of the key points in GR is that mass and energy are interchangeable by E=mc^2. Mass-energy equivalence is how Fission bombs work.
"Relativistic mass" is one of those concepts that was considered, bandied about in the 70s for a while, then ultimately discarded as not helpful.
Photons have no mass, period. An ensemble of photons may have a non-zero invariant mass, but there's just no way in which it's correct to say a photon has a mass.
This has nothing to do with saying that you can't convert mass to energy. Electron-positron pair creation from ultra high energy gamma rays has been a known thing for a long time. Pulsars are thought to run partly off of pair creation off the magnetic field, etc.
Stupid question maybe, but if you can create mass from photons, wouldn't that start to affect the shape of the universe if it happened enough? Could you use this, in theory, to turn an ever expanding universe into one that contracts? Just keep turning energy into mass?
Nope. In GR, energy has as much "gravity" as the equivalent mass, so it won't affect anything. (Energy is more tied to the expanding universe, though, as the wavelength increases too, so it's an important distinction in early-universe cosmology.)
