Massless particle will not experience the passage of time and cannot evolve or change. However, for a system where these massless particles interacts (and the system evolve and change) the passage of time, as well as mass, will emerge. So the system will experience the passage of time (and have...
With regard to increasing of mass, the notion that your mass increases as you approach the speed of light is a bit misleading (or leads to confusion)
From an observer at rest it is appearing that your mass is increasing as you approach the speed of light, because you appear to accelerate less...
In SR, if I see you are moving at half the speed of light (c), I will also see your clock run slower such that you will measure the light moving away from you at the speed of light (c). So don't matter at what speed you are traveling, light will always travel at c.
So at rest, light will travel...