Suppose I traps some free electrons using magnetic field and cool them to subzero temperature, can I get a super-electron this way? and does it violate the exclusion principle?
No because electrons are fermions not bosons.
It is possible for fermions to form a BEC if they can pair up with spins opposed to form a bosonic particle. For example this is what happens in superconductivity and superfluid helium-3. However this requires some attractive force to bind the particles together. In superconductivity deformation of the crystal lattice provides the attractive force while in helium-3 it's the London force.
However for a gas of free electrons there would be no such attractive force to pair up the electrons, so they could not form a BEC.