What happens to electrons when they are detected? I know photons are absorbed and annihilated when they are detected but   
a) when electrons are detected, are they absorbed and annihilated? and    
b) if electrons are ever absorbed, what happens when they are? Are photons omitted?
 A: You have to distinguish the behavior of fermions and bosons. They have different spin quantum numbers, half integer for fermions, integer for bosons.
There is no conservation law for bosons, whereas fermion number is conserved.

Besides this spin characteristic fermions have another specific property: they possess conserved baryon or lepton quantum numbers. Therefore what is usually referred as the spin-statistics relation is in fact a spin-statistics-quantum number relation.2
In contrast to bosons, as a consequence of the Pauli principle only one fermion can occupy a particular quantum state at any given time. If multiple fermions have the same spatial probability distribution, then at least one property of each fermion, such as its spin, must be different. Fermions are usually associated with matter, whereas bosons are generally force carrier particles; although in the current state of particle physics the distinction between the two concepts is unclear.

The electrons are fermions and cannot just disappear the way the photon, a boson, does just balancing energy and momentum. The other quantum numbers have to be balanced also. Thus depending on the interaction under observation, an electron will scatter off the field of molecules and atoms and other free electrons, radiating photons, until its energy is low enough to be bound in a molecular or atomic state, even increasing the molecule's charge. If it is unlucky enough to meet a positron, maybe in the electron positron cloud at the center of our galaxy, it will annihilate to two photons, because the quantum numbers will be balanced.
In vacuum it will just stick around if it has close to zero kinetic energy and is away from molecular surfaces.
Detection, "here is an electron" happens in a detector where a signal is given that an electron has been captured , as a dot on a screen for example, or by the ionizing radiation it leaves in a chamber until its energy degrades enough to be captured by a molecular level.
