Questions about Time, Distance, and Relativity

[Mudhammam]

No doubt this is all going to come out very clumsily, so please bear with me. I'm currently reading Brian Greene's mind-blowing but pleasantly accessible classic, The Elegant Universe. I was reading the section "Motion Through Spacetime" to my girlfriend in which Greene makes the point that a photon emitted at the big bang is the same age as it was "then." My girlfriend, recalling the recent Cosmos series, asked how it is that light can be used to measure the age of the Universe if the speed at which it travels eliminates any passage of time (in reference to the point-of-view of the photon). My muddleheaded answer to her clearly failed to clarify the reasoning for this. That's my first question. But furthermore, it created an additional question-mark for myself. When we speak of the passage of time and distance, per the theory of relativity, are we only speaking in terms of reference to ourselves? That is, is it correct to say that for us as observers, on "our clocks," the Universe is such and such an age, but that on other reference points, it could be different? Is the same true for distances? In other words, does the relativity principle eliminate what may otherwise have seemed to be the sure falsification of (classically conceived) contradictory statements since the truth about an event can only be measured from a particular vantage point, and that among a plurality of positions moving through spacetime each is entitled to a correct but different interpretation of said event, relative to themselves? Does it falsify "objectivity" in regards to the age of the Universe, as that seems to only be true per our level of description, given our state of motion, but not necessarily from a person, say, hypothetically standing on the edge of a black hole's event horizon?