Varsha,
I am no physicist, and my graps of relativity is shaky at best. I'll probably get slammed by my betters for inaccuracies, but perhaps, in a sense, this makes me the best person to answer the initial part of your questions. I say initial because grasping the full meaning of relativity is a process, IMO, not an all-or-nothing issue.
First, the thing about time is that it is a very subjective measure. Common expressions like "Time flys when you're having fun" is a profound thing when you realize that an idividuals perception of time can be very different from someone else's.
Of course, you might argue that the actual passage of time was the same for everyone, so perception was illusory. What then is the point? The point is that it actually works the other way around. A variation in actual time can be perceived by you as "normal" time by your watch. You assume the flow of time is constant, so your perceptions are based on that.
I've read that if we watched a body fall into a Black Hole, we would never see it actually reach its destination, because of time dilation. The flow of time for that body is different to ours. Apparently, proximity a gravity source slows the passage of time. So if you are watching this event from a "safe distance", it would seem to you that the falling object slows down. To you, the object would eventually appear to be just hanging there (if you could still see it).
For the object, time is moving along "as usual". If it was observing you, it would seem that you were speeding up, the whole universe would be "running" faster and faster.
I have not the least notion of why time behaves like this, but this illustrates that time does flow differently depending on your point of view. "Normal time" is that of the observer. "Observer" is a very common word you will encounter in matters of relativity; as I hope I've explained, the point from which an event is observed makes all the difference.
So, to come back to your first quote, I do not agree. To us (the observer), the photon would appear to travel faster than expected because we are in a gravity well (the Earth's), while the photon is relativly free of of gravity. BUT WAIT! Isn't the speed of light a constant? Ah, so what happens is the spectrum of light we see which is the photon is shifted, but we perceive the photon as moving at the "right" speed - c.
For the photon, well now, what happens there? Again, something traveling at c ALWAYS travels at c. Is it not said that anthing traveling at c has infinite mass? IF that is so, then from the photon's perspective it isn't traveling at all, but exists everywhere at once. There isn't any "trip" from the photon's perspective, so discussing how long it takes is a meaningless question - it hasn't gone anywhere since it is everywhere.
For an object moving at less than c, time is an interesting subject to discuss; the object has some velocity, some specific position in the universe, and some time it will take to get where it's going. Our perception, from a relatively stationary platform, deep in a gravity well, will be very different to the perception of the object making the trip, but such a comparison has some meaning.
For the Photon, however, I suspect time is irrelevant. From its point of view, it is everyWHERE, and I am thinking it is also everyWHEN.
Hence, no matter what you do along the lines of chasing after the photon or running away from it, you'd always measure its speed as c, it will just be red- or blue-shifted in your perception of it.
To the Photon, you don't exist by any meaning of the word you or I can understand!
