1) Flyback name. In TV electronics, the Damper or Flyback diode conducted during the time when the transformer was off, causing the beam to "fly back" to the beginning of the next scan line.
From repairfaq.org:
"The term 'flyback' probably originated because the high voltage pulse that charges the CRT capacitance is generated by the collapse of the magnetic field in the core of the transformer during the short retrace period - when the electron beam in the CRT 'flies back' to the start of a new scan line. The flux in the core changes slowly during scan and is abruptly switched in polarity by the HOT turning off during the flyback or retrace period."
Link.
2) MOVs can have a finite life. I agree that in this application, they should last forever if properly sized. Any line spikes that could cause damage would be absorbed by the coil. MOV degradation is a known problem.
Documented here.
3) Peak reverse should be at least 10 times supply voltage. For 24V, the spike can reach 300 volts.
4) Relays have delayed dropout with diodes. Yes, the voltage across the diode will be under one volt, because all of the energy is being sent back to the coil, delaying the collapse of the magnetic field.
From Wiki:
" In an ideal flyback diode selection, one would seek a diode which has very large peak forward current capacity (to handle voltage transients without burning out the diode), low forward voltage drop, and a reverse breakdown voltage suited to the inductor's power supply. Depending on the application and equipment involved, some voltage surges can be upwards of 10 times the voltage of the power source, so it is critical not to underestimate the energy contained within an energized inductor. When used with a DC coil
relay, a flyback diode can cause delayed drop-out of the contacts when power is removed, due to the continued circulation of current in the relay coil and diode. When rapid opening of the contacts is important, a low-value resistor can be placed in series with the diode to help dissipate the coil energy faster, at the expense of higher voltage at the switch."
Link.
Again, 1N4937 is fine. 1 Amp at 600 volts. The Wiki starts confusing the issue by mentioning faster switching Schottky's, but this is related to power supply design, not snubber diodes.
A note on diode construction. The limit is about 400-600 volts. To make a 1000 volt diode, they put 2 in series. That's why the forward conduction jumps from 1V to 1.7V for the 1000 volt (for 1N and UF 400x series)
To revise my earlier statement about first choice, it's UF4004. UF4007 is a 1000V device.