The motion of an electron accelerating in a uniform magnetic field As far as I know, when electrons travel perpendicular to a uniform magnetic field, the Lorentz force makes the electron undergo circular motion. As this electron undergoes circular motion, it emits EM radiation, so it goes through a spiral trajectory with a continuously decreasing orbital radius.


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*As this electron emits EM radiation, its energy should decrease. However, as far as I researched, the velocity of the electron should remain constant as the magnetic field does no work on the electron. Does this mean that the loss of the kinetic energy of the electron comes from its loss of mass? If it does, what is the formula for the mass of an electron (or any charged particles) that is accelerating in a uniform magnetic field and thus emitting EM radiation?

*Where can I find some videos that actually show the spiral trajectory of an electron?

*How fast should an electron be moving or strong the magnetic field be for the electron to undergo a spiral path like the one shown below?

 A: 
As this electron undergoes circular motion, it emits EM radiation, so it goes through a spiral trajectory with a continuously decreasing orbital radius.

What is electromagnetic radiation at the particle level of an electron ( a quantum mechanical particle)? The emission of photons. The probability of emitting a photon which will take away energy and thus reduce the energy of the circling electron, can be calculated given the initial values and the magnetic field. 

the velocity of the electron should remain constant as the magnetic field does no work on the electron

The velocity is reduced according to the energy taken away from the photon.  This is called synchrotron radiation and is important for high energy particles.
The spirals you see in the picture you include are not due to this loss of energy, which is very small at the dimensions  (momentum)of the circle shown, but due to the electromagnetic interactions of the electron scattering off the electrons of the material of the bubble chamber. These scatters make the little dots which make the trajectory visible. It is called energy loss by ionisation.. It is used in particle detectors together with the trajectory measurements to determine the mass of the particle.
In addition, spirals happen also when there is an angle with the magnetic field, so one cannot tell from one picture  unless a second one of the same event is obtained so as to calculate the angle to the magnetic field the electron initially makes.
As for your question 2. 
These are very fast moving electrons and cannot be caught in a video.
For 3.For example in the 2 meter bubble chamber at CERN that has similar pictures the magnetic field was 1.5 T. 
A: The constant speed of electron is by assuming electron does not emit EM radiation.You just mixed up with it.
A: The whole question is perfectly formulated, but allow me to reconsider these formulations nevertheless. 

As far as I know, when electrons travel perpendicular to a uniform magnetic field, the Lorentz force makes the electron undergo circular motion.

The Lorentz force is the expression for the observed phenomenon that moving charges are deflected under the influence of a magnetic field (moving or not in relation to the charge, but necessarily not parallel to the trajectory of the charge). The Lorentz force is the result of the deflection of charges by magnetic fields and not the reason for the deflection.  
Such an approach leads us to the question, how the magnetic field and a moving charge interact. With your next sentence you are close. You describe in detail what happens in the interaction.

As this electron undergoes circular motion, it emits EM radiation, so it goes through a spiral trajectory with a continuously decreasing orbital radius.

In addition to its electrical charge, the electron also has its own magnetic field. The electrons are magnetic dipoles. Now an interaction between the arriving electrons and the external magnetic field occurs naturally. The external field aligns these dipoles parallel to its own field.
The uncertainty in the description of what causes the Lorentz force (the deflection of the charges) is as follows.
Does the alignment of the magnetic dipole of the electron cause its deflection (gyroscope effect) and the lateral acceleration causes a photon to be emitted?
Or does the alignment of the magnetic dipole of the electron cause the emission of a photon and the electron is deflected by its recoil moment?
In any case, the relation between photon emission and loss of kinetic energy is obvious.

As this electron emits EM radiation, its energy should decrease.

It happens this way.

However, as far as I researched, the velocity of the electron should remain constant as the magnetic field does no work on the electron.

The magnetic field is like a catalyst or spring. It interacts again and again with the electrons magnetic dipoles. However, these electrons escape permanent alignment by photon emission.

Does this mean that the loss of the kinetic energy of the electron comes from its loss of mass?

If you apply the equivalence principle between energy and mass, then yes. But it would suffice to say that the loss of kinetic energy results from the emission and recoil of photons.
