In this figure from Wikipedia, we know that electron's wavefunction collapse at screen F, causing an interference pattern. Does it mean that in this case when the wavefront arrives at the screen, the screen does something similar as a 'measurement' to cause the wavefunction to collapse? If so, why wouldn't the electron's wavefunction collapse at screen S2 which will produce no interference pattern at the screen F, since the wavefront first arrives at the screen between slits B and C?
... we know that electron's wavefunction collapse at screen F, causing an interference pattern
No, interference does not require collapse. The screen with the slits effectively modulates the electron wave function with a function that produces the interference pattern in the far field. What then happens is that the absorption of the electron at the screen causes it to be localized at a specific location with a probability distribution given by the interference pattern. The localization of the electron at the point where it is absorbed does not require collapse, as understood in the Copenhagen interpretation of QM. All that is necessary is that the measurement basis of the absorption process implies a localization. Such a measurement basis is determined by the atoms in the screen that would absorb the electron.
This is an insightful question. Actually, the comment by @my2cts is right on the mark.
"Wave function collapse" really doesn't happen. Instead, the microscopic quantum state (location of photon's impact on the screen, which is quantum mechanically indeterminate) gets correlated to a macroscopic state (your perception of the location of the photon's impact on the screen). Your perception state, too, is quantum mechanically indeterminate. What you see is one randomly selected location out of the entire range of possible locations (with the randomness weighted according to the squared amplitude of the wavefunction at each location). If the wavefunction of the whole system is expanded to include you, there will be an infinite number of yous, each seeing the photon apparently hitting a different location on the screen.
As you can see from the comments, wavefunction collapse is just one of the interpretations (Coppenhagen) in QM, actually a very interesting one.
Now to your question, why does the electron's wavefunction not collapse at the first screen?
The answer is the two little slits. As the wave reaches the first screen, you say that the wavefront reaches the screen first between the slits. This implies that you think of the wave as propagating in the displayed fashion.
In reality, the electron as it propagates, a QM object, its trajectory is undefined. It takes all possible paths. Yes QM is a tricky beast and it is very unintuitive to imagine.
What really happens is that the electron as it propagates, reaches the first screen and continues to propagate through the slits, that is why there is no decoherence with the environment, that would cause the electron's superposition to reduce to an eigenstate, with a certain eigenvalue (the position of the electron on the screen).
The photons do not have a well defined trajectory. The diagram shows them as if they were little balls travelling along a well defined path, however the photons are delocalised and don't have a specific position or direction of motion. The photon is basically a fuzzy sphere expanding away from the source and overlapping both slits. That's why it goes through both slits. The photon position is only well defined when we interact with it and collapse its wave function. This interaction would normally be with the detector.
Shooting a single photon through a double slit
What confuses you is that you try to imagine the wave as reaching the part of the screen between the slits first, then decohere, and cause the collapse of the wavefunction. What really happens is that the electron takes all paths and finds a way through the slits. Yes it is very hard to understand how the wave takes all the paths, and knows not to collapse because there are two slits to go through. This is QM.
The most intuitive way of looking at interference that I have encountered is Feynmann's Path Integral Formulation. Loosely speaking, if you have a photon (or anything, really) in location A and want to work out its chance of moving to B, you imagine it taking every possible path between the two at the same time.
How two photons interfere in a double slit experiment
When the electron reaches the second screen, it does not find any slits to propagate through, and finally interacts with the screen, leaving a dot on the screen. That is what you call the collapse of the wavefunction. Its position becomes localized.
In this figure from Wikipedia, we know that electron's wavefunction collapse at screen F, causing an interference pattern. Does it mean that in this case when the wavefront arrives at the screen, the screen does something similar as a 'measurement' to cause the wavefunction to collapse? If so, why wouldn't the electron's wavefunction collapse at screen S2 which will produce no interference pattern at the screen F, since the wavefront first arrives at the screen between slits B and C?
The interference pattern is not caused by the collapse, but by the evolution described by the Schroedinger equation.
Only the fact that the pattern is made of many small spots instead of continuous variation of some measurable quantity requires us to introduce additional modification of the psi function, the so-called collapse of the psi function. This process is not assumed to be described by the Schroedinger equation; instead it introduced by us to take into account the fact that after the record is created, it is certain that the electron was at a definite small spot of the screen at the time of impact.
If we observed records of electrons on the screen S2, we would assume collapse happened there as well. But usually there is no such sensitive recording screen at S2 in this kind of experiment, so there are no records, so we do not assume collapse there.
Quanta are combinations of energy, momentum, angular momentum and charges that get exchanged irreversibly between quantum fields (in this case the quantum field of electrons) and external systems (in this case the screen).
The important piece here is the word "irreversible". We are removing the electron energy (in classical term this would be the kinetic energy of the electrons) from the quantum field at the screen, which is different from the process at the slit openings, which does not remove energy from the field (and is therefor reversible).
The collapse meme is a very unfortunate one. It should be banished once and for all from the terminology of physics, if for no other reason than that is has absolutely no explanatory value. In the Copenhagen interpretation the distinction between reversible processes that do not remove the energy of the quantum from the system (often represented by the Schroedinger equation) and irreversible processes (represented by the Born rule) is made extremely obvious. This aspect of the structure of the conventional theory is often not sufficiently emphasized in the classroom, which seems to leave a lot of students wondering why there is such an "artificial" divide between the two cases. It is obviously not artificial, at all, but an experimentally verifiable feature of nature: energy either stays in the quantum system or it gets removed by the measurement.
