Why do fermions not have a Bose-Einstein Condensate phase? My misconception was that electrons were present in a BEC. However, how do fermions behave in low temperature and how does it differ from the behaviour of bosons?
 A: Electron are spin 1/2 particles, and hence are fermions.
Fermions do not have a Bose-Einstein condensate phase (that is, as the name suggests, for bosons).
In some particular situation (low temperature etc) they can give rise to a Bose-Einstain condensate if they couple and form  bosonic two-electron states (Cooper pairs) but then it is really the couples that form a condensate, rather than the electrons
Addendum:
In general (an other example are alpha particles, that are bosons but made of nucleons that are fermions) as long as you are in conditions such that you can ignore the internal structure of the bosonic composite you can have a Bose-Einstein condensate.
Addendum II because the OP changed radically the question:
Electrons are fermions; fermions have antisymmetric wavefunction, that is, if the system have discrete energy levels, thy are subjected to Pauli exclusion principle, and therefore two electrons cannot occupy the same level.
Bosons, on the other hand, can "collapse" to the lowest energy level and this is the BE condensate
