Re: Why does the magnetic field of an electron reverse direction?

It seems you are really talking about two different things so…
First: Let’s clear up, I hope, a little confusion.  Moving any charged
particles will produce a magnetic field around them.  If we move these
particles across an external magnetic field the two fields interact and the
particles feel a force mutually perpendicular to the external field and
their direction of motion.  This is the force that we say induces a current
or voltage and it has nothing to do with whether or not the particles in
question (electrons in the case of wires) each have an  inherent individual
magnetic field of their own or not.  Most high school and freshman college
physics textbooks will have some pretty good pictures of this behavior.  At
the atomic level in a material like copper there is about one electron per
atom that is essentially free to move about in the wire.  If this wire is
moved across an outside magnetic field then these electrons are seen as
charges moving across that field and are pushed in one direction or the
other by the interaction described above.  What reverses direction is the
force acting on them not their own individual magnetic fields. Since copper
is a conductor in which the electrons are free to move then they will try
to do so resulting in an induced current or voltage along the direction in
which they are forced by the external field.
Second:  Not all charged particles have inherent magnetic fields but it is
true that electrons do.  So do protons for that matter.  You can imagine
them as tiny bar magnets (dipoles).  When you read about flipping,
reversing, or aligning these fields you are most likely discussing either
the formation of magnetic domains as in ferromagnetism, (here again the
physics text will come in handy with some good pictures) or some kind of
magnetic resonance process, Electron Spin Resonance (ESR), Nuclear Magnetic
Resonance (NMR), or Magnetic Resonance Imaging (MRI).  These last three and
their relatives involve trying to line up some of the individual electron,
(or proton), fields with an external field and then trying to flip them
over with an outside input of electromagnetic energy (usually radio or
microwave). Exactly which frequencies are required tell us about the atoms
with which these electrons interact so you are right in the sense that the
net field of the atom is important too.  At ordinary temperatures and
typical field strengths random thermal motion will keep the majority of
these particles jumbled with respect to the external field so we don’t
notice this affecting what we discussed in the first part of this answer. 
With a strong enough field however and sensitive enough detectors we can
measure the effect for the second case and it turns out to be a very
important tool for science and medicine.