I'm wondering why matter and antimatter actually annihilates if they come into contact. What exactly happens? Is that a known process? Is it just because of their different charges? Then what about neutrons and anti-neutrons? I guess that would be about their quarks, but wouldn't it be very unlikely what all of their individual quarks would actually touch the antí-quarks of the other (anti)neutron at exactly the same time?
The QED Lagrangian contains the fundamental vertex:
[Image source: wikimedia]
Depending on how you view this diagram, it can represent different things. If you consider time flowing vertically upward, then it shows an electron emitting a photon as it accelerates. If instead, you consider time flowing to the right, it looks like pair production from a single photon. Time flowing to the left would look like annihilation into a single photon. However, in these last two cases, if you try to conserve momentum with this diagram alone it is not possible. In the center-of-mass frame, the colliding electron and positron system has vanishing total momentum. The photon, however, has momentum in all reference frames.
So, it is necessary to create two photons in order for annihilation to occur. This can be done as follows:
[Image source: wikimedia]
This is the process by which annihilation occurs.
If you replace the electron with any charged fundamental particle, the above is also true. For neutral fundamental particles, annihilation occurs via the weak interaction, where the vertex involves a Z boson instead of a photon.
The answer is unknown. However, we have a good model, QED, that mathematically describes a continuous process of particle creation and destruction.
The QED Hamiltonian, the generator of infinitesimal time evolution, has an interaction term that creates/destroys electrons, positrons and photons. This happens at a point, an event, in spacetime. This "vertex" involves three particles one of which must appear either before or after the interaction.
For example, there might only be an electron before but after, an electron and a photon. But, there is a problem: momentum and energy cannot be simultaneously conserved; at least one particle must be "off-shell" or "virtual".
However, the virtual particle can propagate to and be involved in another vertex and the combination of the two can conserve momentum and energy.
So, for example, consider a vertex with an electron before and after, a real photon and a virtual electron. Consider another vertex with the virtual electron and a positron before and after, a real photon.
The combination of the two processes is "seen" as an electron and positron before and after, two photons; two matter particles are destroyed and two vector particles are created.