Does the photon itself have its anti-particle?
Yes, the photon has an anti-particle... itself :-)
If our universe was made mostly of anti-matter, instead of matter, would the laws of physics be any different?
That is a very interesting question.
My gut instinct would be to say yes: If they were identical for matter and antimatter, that would mean a perfect symmetry between matter and antimatter, which is obviously not observed.
However, if you imagine that all matter in the universe changed to antimatter in an instant, I am not sure you would notice a difference in your daily life. The consequence of differences in physics laws would be unnoticeable on a human timescale. Only over an extensive amount of time, maybe eons, the slight assymetry would maybe show up as a gradual increase of background gamma radiations over the ages.
You can also say that the terms matter and antimatter are relative. The inhabitants of an universe where antimatter is favored would call it matter, because that's what they are made of, and vice versa (our matter would be their antimatter). ;-)