No, photons do not create a gravitational field.
As for why moving charges produce a magnetic field, there is no "reason" for this--it's just a law that they do.
Here's a more interesting question:
You have two observers. One is standing still and watches a charged particle fly by. Since he knows moving charged particles create a magnetic field, he predicts what the magnetic field will be (and is correct). The other observer moves with the charged particle--so to him, the charged particle is stationary. So does the moving observer predict and/or measure a magnetic field?
Edit:
@Andrew Smith: General Relativity explains why light can have no mass and yet still be affected by gravity. This is because mass warps space time, therefore light is merely traveling along geodesics (shortest paths). For flat space this would be in perfectly straight lines, however, if space is curved (due to mass warping it), then it will appear to curve or bend. This is consistent with the idea that photons have no mass yet are still affected by gravity.
Furthermore, special relativity, shows that mathematically, the only possibly way for something to travel at the "speed of light" is if it has no mass. I will offer a caveat though, which is that if light has, perhaps a TINY bit of mass, it still might be possible to go at say, 99.9999999999999% of the speed of light, which perhaps we cannot measure the difference (which might be a bit circular anyway--since we define the speed of light as the speed at which photons travel to begin with).