Multiple interactions can happen.
First, lets look at the "simple" case of an electron interacting with a single atom. There are three options. (1) the electron does not interact with the (neutral) atom - it misses completely and keeps on going. (2) the electron may interact with the electron cloud of the atom. This may lead to a scattering event, kicking a bound electron into a higher energy level, or knocking it off the atom entirely (generating the secondary electrons used for imaging in an SEM). The atom can then relax, possibly releasing an x-ray or an Auger electron, both of which are used for composition analysis of materials. The incident electron is deflected from the original path because of the scattering event. (3) the electron may interact with the nucleus and undergo Rutherford scattering (yup, same formula applies).
The more complex case of an electron impinging on a crystal adds the periodic potential of the crystal. All of the events listed above apply, but in addition the periodic potential allows for diffraction, where the electron scatters off the Bloch waves of the crystal. In a TEM diffraction can be imaged from the transmitted beam, in an SEM it can be imaged from the back-scattered beam (EBSD).
So, pretty much everything you mentioned can and does happen. All are used for various measurements in materials science and physics.