I have a few questions about EOL protection:

-On some ballasts EOL protection kicks in if an arc can't strike within a few seconds, why is that?

-On some other ballasts, EOL protection takes a few minutes to kick in; the EOL tube will go EOL for a little bit, and then the ballast seems to do something and the tube kind of dies down until it's  either glowing very dimly or doing nothing, why is that?

-What exactly is the EOL protection circuit? Does it just simpily cut out a bad lamp?

-Why do some ballasts have/not have protection?

It depends what is the motivation to use the protection in the first place.
1. Prevent annoying erratic operation (lights permanently flashing, cycling, abnormal color shift,...): These leave the system alone attempting to run the lamp for some time and if the erratic behavior continues, shut it down so the failing lamp stop become annoying.
2. Protect the gear from excessive wear and/or damage caused by failing lamp: Generating starting high voltage usually means quite high stress on many insulation, dielectrics and component involved in generating that voltage. Normally the design assumes it remains in this stressing mode for only very short time (then the lamp starts), so have very limited time they can withstand it.
3. Protecting from failing lamp becoming a hazard (explosion, fire,...). An example could be fire from overheating fluorescent electrode assemblies, HID explosions,...