It's not obvious what you mean by "a time dilation effect infinite in both directions". Did you mean into the past as well as into the future?
A massless particle experiences no flow of time; not into the future and not from the past. Just to make things even stranger it experiences no distance either i.e. as far as a photon is concerned there is infinite time dilation and infinite length contraction.
Re the second paragraph you've edited: I'm still not sure I understand you. Are you asking if, for example, the present and the beginning of the universe appear to the particle to be at the same moment, even though the particle hasn't existed since the beginning of the universe?
If so, I suppose there is a sense in which the answer is yes, but this answer would be very misleading. A massless particle doesn't experience anything so it's meaningless to say that for the particle some event A happens at the same time as some event B. In effect a massless particle has no time dimension, so it's meaningless to talk about whether two events happened at the same time.