Do photons leave a permanent trail? Do we analyze this to know from where they come?

No, they don't leave a permanent trail. We use clever techniques of bending and reflecting their paths to determine which direction they came from.

The first principle is this: Photons usually travel in pretty predictable paths, and mostly in straight lines. We have tons and tons of everyday, practical experience to verify this. If it weren't true, then you wouldn't even be able to make your way visually from room to room. There are exceptions, of course. Things like fog will scatter photons in random, unpredictable directions. But that's when we turn off our detecting instruments and wait for the fog to clear.

Consider a very distant source of photons, like a galaxy emitting radio waves. If we have simple detector like a car antenna, it may be able to "detect" the radio photons from that galaxy, but it can't tell which direction they're coming from. However, if we have a big parabolic reflector, that's a different story. a paraboloid is specifically shaped so that any photons coming in parallel to its main axis will bounce off in such a direction that they'll all focus at a particular point along the axis. So then we stick a photon detector right at that point. Finally, we move the whole rig until we pick up a strong signal on the photon detector. When that happens, we know we're pointed at the radio photon source.

Optical telescopes work the same way. They typically have a parabolic mirror at the back end that reflects any (visible light) photons coming down the tube. If the tube is pointed in the right direction, those photons will reflect off the mirror in just the right direction to form an image in the eyepiece. If the tube's direction is just a little off, the photons will either miss the mirror entirely, or they'll reflect at the wrong angle and miss the eyepiece. So, if you see an image in the eyepiece, It's a safe bet that the direction the tube is pointed is the direction that the photons came from.

Some (usually smaller) telescopes use lenses in the front instead of mirrors in the back. In this case, the lens bends the incoming photons at very particular angles, making them all focus to form an image at the eyepiece in the back. Because of how lenses work, if the tube is not pointed in exactly the right direction, the place where the photons focus will be off-center and will miss the eyepiece, and you won't see the image. Again, if you see an image, you can bet that the photons came from the direction your telescope is pointed.

For that matter, the lens in our eye is a pretty good tool too, for helping us "analyze where photons came from". The car keys on my dresser send out photons in all directions. Some of those photons hit the lens in my eye. Those photons are competing with photons from a lot of other places in the bedroom too, which are simultaneously entering my eye. But because of the way lenses work, the particular photons from my keys get bent (refracted) in just the right way so that they land on a particular spot on my retina, away from all the other photons. My brain is able to interpret this, so I can turn my eyes to "look at" my keys. I have figured out where those photons came from. I can now drive to work.

Photons don't leave trails. If they have passed by us - the only way to know is if they hit another object - and reflect back to us. Otherwise, we have no way of knowing about photons having 'passed by' us.

A photon is a particle and does not "leave a trail".
A trail is something left behind and this can not apply to a photon. If it interacts it ceases to exist on absorption, if it does not interact there is no trail.

No photons don't leave a permanent trail. No we can't analyse that trail to find out where the photons come from.

Photons however travel in a straight line which is how we know where they came from. Bear in mind though we don't usually detect single photons but entire streams of photons.

Paul shot himself in the foot. A stream of phones leaves a trail, that's what streams are, a trail of photons.
Light travels in a straight line EXCEPT they are constantly bent by gravitationally strong planets and stars.
See how easily it is to prove Paul wrong.
The ignorant love proffering their Bull Shiite.