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From what I've so far understood about light, a photon is emitted somewhere and after some time it's absorbed somewhere else.
Have we had experiments that confirm the path taken or something akin to verifying that photon really do "travel"? If so what are they..
The central point of the question is somewhat ambiguous, but here is an effort to answer it. I am sorry in advance if I have misunderstood it.
Does light/photons travel?
The question whether light travels from place A to place B or not, can be answered mainly by experience and experiment/observation. When you hold a torch in the dark and you aim it at some point in the background where it is dark, you can see its effects almost immediately. From having being dark, now it is bright and you can see the objects that exist there. That means that light not only travelled there and illuminated the area, it also came back to your eye to give you the information about the objects. This means that light has not always been there, suspended in the air, waiting for you to turn the torch on and make it become reality. I don’t think this is how you envision it.
Does light “feel” the existence of space?
This type of questions touch on the borders of ontology, somewhat. It is not very easy to formulate answers because one has to talk in terms of metaphysical notions and concepts which, unfortunately, fall outside the scientific method of thinking. But let us take a look at it from this point of view: Imagine we send a laser beam from one side of our room to the other. Watching it without an apparatus it looks as if light did not have to travel at all, it looks as if the event evolved instantly. A very sensitive apparatus, however, can sense that light has actually taken some time to go there and back. The situation can become more obvious if we try to send the laser beam to the moon and back (this has been done.) Even we, without any apparatus, can tell that the distance involved must be huge. So space becomes important and even light “feels” the vastness of it. In the experiments you mentioned, the extremely sensitive detectors can distinguish photons arriving with a time difference just a few nanoseconds or less, due to the slightly different paths they take (space becomes very important in less obvious ways) Light can even “feel” the geometry of space-time, as is demonstrated by the deflection of light-rays passing near the surface of the sun, during a total solar eclipse. Light can “feel” the immense density of a Bose-Einstein condensate by slowing down to incredibly low speed. You can run fast enough and catch up with it!!
The question whether or not light takes a well defined path to go from A to B involves quantum mechanics, and from your comment I read that it does not interest you at the moment(?)
You should have a look at How does a photon travel through glass?. This is really a duplicate of your question but I haven't voted to close your question because I'm guessing you're asking for a fairly basic answer.
You're thinking of the photon as little bullets fired from some point and hitting some other point, but this is only a partial description of light. Light is a quantum field and may interact as a particle or interact as a wave under different conditions. The question "what path does the photon take" can't be answered because the light isn't particle and therefore doesn't have a well defined path.
krs013 mentions the double slit experiment, and this is a good example because the light travels through both slits at the same time. It obviously couldn't do this if it was behaving as a particle.
