Does something change in a magnet when an electron crosses its lines? An electron can cross the magnetic field of a steady magnet and flip its spin. What changes in the body of the magnet? Does some electron inside flip its spin opposite to that of the passing electron and if so, does it diminish the magnetic field of this magnet?
 A: 
An electron can cross the magnetic field of a steady magnet and flip its spin. What changes in the body of the magnet?

In permanent magnets a remarkable part of the subatomic particles is aligned with their magnetic dipoles and form the macroscopic magnetic field.
If you move another permanent magnet towards the first one, the alignment of electrons is intensified in both magnets. Strictly speaking, the common magnetic field of both magnets is stronger than the sum of the fields before the approach. This has to do with the - in sum better - alignment of the electrons to the common magnetic field of the magnet.
Secondly, when the magnets have moved away again, the additionally aligned electrons - which are trapped in their atomic and molecular bonds - are pushed back into their old orientations. The magnetic fields of the two magnets remain at the same strength as before.
What does this have to do with your topic. A single electron has a intrinsic magnetic dipole, it is a tiny magnet. Probably the electrons diplle is oriented in random direction and corssing the permanent magnet, it gets aligned. On the over hand it „helps“ the electrons inside the magnetic material, to get aligned stronger in the common direction of the macroscopic magnetic field.

Does some electron inside flip its spin opposite to that of the passing electron and if so, does it diminish the magnetic field of this magnet?

From the above said it is clear that usually nothing flips because of the strong atomic and molecular bonds. In sum the orientation of the electrons show more in the direction of the common field. The magnetic field amplifyes.
From your comments:

... the electron is strictly regarded as pointlike and there is no rotation when its spin changes. It flips from say -1/2 to +1/2. I am not even sure that it changes spin z-component or is just projected on -1/2 or +1/2. But then also all electrons inside the magnet must be projected in regard to the electrons magnetic field.

The numbers you mentioned are introduced to show the orientation of electrons in atoms. For example, the two electrons in Helium have an antiparallel orientation of their magnetic dipoles, expressed by -1/2 and +1/2. And for a free heliom atom both electrons together can be spatial oriented randomly.
In a aggregation the termic movement of the atoms and molecules prevents the self-alignment of teh magnetic dipoles. Under some circumstances - near 0 Kelvin and not as a solid as a starting point - the self-alignemnt works and you get a BEC.
So it is not enough of a single electron, crossing a material, to influence all the electrons to „be projected in regard to the electrons magnetc field“.
A: The magnet generates a magnetic field $\mathbf{B}_{\text{magnet}}$. The electron has a dipole moment $\mathbf{m}_{\text{electron}}$ due to its spin, which will experience a torque $\boldsymbol{\tau}$:
$$ \boldsymbol{\tau}_{\text{on electron}} = \mathbf{m}_{\text{electron}} \times \mathbf{B}_{\text{magnet}}. $$
By Newton's third law, the electron itself will also exert a torque on the whole magnet:
$$ \boldsymbol{\tau}_{\text{on magnet}} = \mathbf{M}_{\text{magnet}} \times \mathbf{B}_{\text{electron}}, $$
where $\mathbf{B}_{\text{electron}}$ is the field generated by the electron's spin, and $\mathbf{M}_{\text{magnet}}$ is the magnetic dipole moment of the whole magnet.
Torque goes as inertia, which goes as mass. Since the magnet is probably much more massive than the electron, the torque on the magnet is negligible. The only visible effet is then the electron flipping its orientation until its spin is parallel to the magnetic field of the magnet.
