What is the mechanism by which an electron and a positron annihilate? An electron and a positron annihilate in pure energy.
What is the mechanism by which an electron and a positron annihilate each other, losing their mass and their electric and magnetic field? Especially, does the charge and by this the fields vanish instantaneous or during some time period? The same for the mass.
 A: The main mechanism is the scattering process $e^-+e^+\to 2\gamma$ admitted by quantum electrodynamics (QED).. Higher order process also contribute but much less. The process is not instantaneous. Since it is calculated as a scattering process, it is defined strictly speaking between times $-\infty$ and $+\infty$, but for practical purposes - like in any scattering calculation - any mesoscopic time scale suffices.
A: You say in a comment

Im not enough educated for QFT.

but you ask

An electron and a positron annihilate in pure energy.

An electron and a positron are quantum mechanical entities, and their behavior can only be described using quantum mechanics, and special relativity, in addition to momentum energy and angular momentum absolute conservation laws.

What is the mechanism by which an electron and a positron annihilate each other,

When a particle (electron in this case) and its antiparticle(positron) trajectories meet in space time, by definition all the quantum numbers may add up to zero within a volume in phase space where there is a quantum mechanical  probability for the pure energy of the two particles to manifest in possible new particles that also add up to zero quantum numbers. This probability can be calculated using Quantum Electrodynamics, QED,

losing their mass and their electric and magnetic field?

The probability can be calculated using QED, yes , the charges and magnetic moments also have to  add up to zero for the probability of the annihilation to be measurable in a phase space volume.

Especially, does the charge and by this the fields vanish instantaneous or during some time period? The same for the mass.

Again this is a quantum mechanical interaction controlled by quantum mechanical probabilities that can be calculated using QED. There will be an interval of time delta(t) where the probability of annihilation is measurable, depending on the experimental accuracies.
