Wikipedia about double slit experiment and my question about that

The fringes will appear where the third polarizer has had its effect on the light from both slits. Elsewhere the screen will appear uniformly illuminated (no fringes). So you get a region with fringes and a region with no fringes.

The classical physics explanation can help here. When the electromagnetic waves at the slits are polarized in orthogonal directions, then when the waves overlap and thus interfere downstream of the slits, the result is a region filled with electromagnetic waves whose intensity is independent of position but whose polarization does depend on position. If these waves then hit a screen which glows by an amount proportional to intensity but independent of polarization, then one will see no fringes. But the fringes are still there in the electromagnetic radiation incident on the screen, in the pattern of polarization. For example, from one fringe to the next the polarization changes from +45 degree linear to left circular to -45 degree linear to right circular then back to vertical. By using a 45 degree polarization filter one absorbs the -45 degree regions and transmits the +45 degree regions, so then the fringes are seen on the screen. There is no need to invoke the language of delayed choice to explain this phenomenon. Indeed one does not need quantum physics at all. For ordinary light sources classical electromagnetism here gives a correct treatment.

If one did the experiment with single photons then one would need quantum physics. In this case the argument above still largely applies, but now it applies to the quantum wavefunctions, including the photon spin degree of freedom. The spin state and the position state become entangled in such a way that the spin direction of the photon becomes a function of position. The absorption at the third polarizer then converts this into a variation of intensity.

Yes, if the third polariser is near the screen, photons which pass through it will produce and interference pattern, while those which do not will not produce an interference pattern. I have not done the experiment, but this is the theoretical prediction.

I would not describe "which path" information, however. Quantum mechanics calculates probabilities for where the photons will be detected in the given experimental set up. The reason it works is that classical concepts, like position, only make physical sense when they are measured.

This is because position exists only as a relationship with other matter. Position in space does not make sense. Even in the classical world we can only say where something is if we say where it is relative to something else; we cannot say where it is in space. In the macroscopic world objects are in continuous interaction with their environment, and always have position, but if a quantum particle has too few interactions to generate the property of position, then it has no position.

If the concept of position does not make sense, then nor does the concept of path. All we can do is calculate probabilities for where photons will be detected; we cannot say where they are when not detected.