As an analogy, consider the photon that strike your face and reach my eyes, we say that that photon carries information about your face which then helps me to identify you,
You are confusing the individual photons with the electromagnetic wave that is light, which is composed out of a zillion photons. It is the superposition of the wavefunctions of individual photons that can carry information.
but don't these photons collide midway with air molecules, and if they still somehow retain the information,
An individual photon can be absorbed or scattered out of the beam it is composing. A few photons scattered out of the beam that is carrying information make no difference as the number of photons is enormous.
If a photon is scattered coherently with the other zillion photons the beam itself carries information of the way all these photons are building it up, in the total wave function, which is a superposition. That is how images can be refracted and reflected.
At some point in opaque and non reflecting surfaces the coherence is lost and any information the beam has been carrying is lost.
then going by the same logic why don't I see the image of things that photon collided before coming to your face like fans, bridge etc
The individual photon is distinguished by its energy and direction, +1 or-1 spin projection on its momentum. When you see an image, you see the collective superposed interaction of the beam of photons, light. When light strikes a "fan, a bridge etc." it disperses and carries information from the object. The part of the light reflected from the bridge when it hits a second object loses the previous coherence by striking and reflecting back, and coheres in a new image pattern.
Generally coherence is lost as electromagnetic beams interact with matter.
In cosmology it's frequently said the photon from the early universe carry information of that time.
The experiments measuring cosmic microwave background radiation are very careful to look at clear regions of the sky. Some are on satellites for this reason so as not to have interference from backgrounds. The photons they are measuring have not interacted before interacting with the detectors of the experiment. If they had been absorbed they would not have arrived at the detector.
The individual photons carry information of their directions, which way the were coming, and their energy, which was absorbed to give the count. That is how the CMB maps are made.
The ensemble of CMB photons arriving at the detectors may have interacted with cosmic dust, for example, or been deflected by strong gravitational fields, and this will affect the polarization displayed by the "beam" of arriving photons. This polarization has been measured in the Planck experiment and with BICEP2 and it is a research question whether it is an original polarization from the inflationary period due to gravitational waves, or due to scatterings from dust. This information is a collective information carried by the beam the CMB photons build up.
But don't they also carry the data afterwards and how do we then differentiate between the data's.
So some information can be carried by the ensemble of photons, as described above.
"The wave is actually probability in the sense that it assigns probability to the space coordinates of detecting photon at a certain time."
No, the emergence of the classical EM wave from the quantum wavefunction of the photon is not trivial, because a classical EM wave is made up of many photons. In particular, it is not the case that the classical EM wave is the wavefunction of a photon. (Even more particular, it is difficult to even speak of the wavefunction of a photon, since photons usually arise in a quantum field theoretic ("second-quantized") description where the notion of wavefunction does not exist (but is, if you insist on something comparable, replaced by a wavefunctional))
Also, do not speak of "the electron wave". While an electron - like all quantum objects - carries wave-like properties and can be described by a wavefunction (which is not a function as you might imagine it if we incorporate its spin in the sense that it does not take values in the real or complex numbers), it isn't a wave in any classical sense, and also still carries particle-like properties. It is a quantum object, neither fully wave nor fully particle at any time.
Nevertheless, all quantum objects (and hence all things you might decsribe as "matter waves") of course carry energy - the energy that is in their rest mass and the energy that is in their momentum, though the energy of any given quantum state may not be well-defined, but "smeared out" over a range of energies.
To ask how physical objects carry the properties they do is, deep down, not sensible. How does a classical particle carry momentum? By having mass and velocity! But how does it carry mass and velocity? By...um...moving and stuff. How does it move? Um... You get the idea. "Why/How" is a question that can be asked infinitely many times, but only answered finitely many before you hit a point where the only answer is because it seems that way.
Best Answer
You talk about light as if it were a person carrying a clip board writing down things on its way to you. It is a physical phenomenon that gets affected as it propagates.
Depending on the various processes that it goes through before it reaches your eye, its amplitude,polarisation, frequency (or wavelength), pulse time etc. get affected from which we can infer what it must have gone through and get to know of the object it must have reflected off or gone through or originated from.
If the frequency is changed, the photon is said to have a different energy from $E=h\nu$. Since light has both particle and wavelike properties, depending on the situation we are in, we can equally talk about $k=\frac{2 \pi \nu}{c}$
Consider these examples:
Say you have a pocket laser source. You shine it on two walls, one at 500 m and one at 1 km. The light travels for more time to get back to you from the second wall. Here, the light is unaffected but only the time is recorded. If you did not know the distance the walls were from you, now you can calculate the distance the walls are at. This is information
Leaves are green. This means that they reflect green light and absorb all the other colours that are present in the sunlight. When you go outside and can "see" a leaf, it is information. Now, the frequencies of a light have been partly affected.
You see stars at night. Light has travelled for many years and the photons have hit your eye. Now you know how the star looked some few years ago. (the light from the nearest star takes around 4.5 years to come to you). Thus, information on the star's position is being carried, along with it's temperature. The wavelength of the light reaching you is carrying information.
Light from objects are also "doppler shifted" : the police use this effect to get the information - the speed of the car that they shine the radar gun on. The frequency is actually changed in this process. This frequency change is carrying information.