How photons travel with speed of light having negligible mass if speed increases mass should also?

Yes, as speed increases towards the speed of light the mass should increase, but only if they have mass in the first place, photons don't have mass which is why they are able to travel at the speed of light (they are light!).
As for exceeding the speed of light, it is a constant for a reason, there is no breaking it, it is a limit only broken in science fiction.