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,...

This depends on which ballast/lamp system we are talking about.

It is problematic to implement a comprehensive EOL protect circuit in a magnetic ballast. Usually a bimetallic switch (if any at all) is placed inside the coil to limit maximum temperature. Thermal time constant for its trip may be in some minutes to tens of minutes.

Starters used with fluorescents on magnetic ballasts may have a thermal bimetallic cut-off built-in.
Electronic starters for fluorescents employ some ignition timeout too, and usually a one-time thermal fuse.
Electronic ignitors for HIDs on magnetics sometimes add a time out circuit for the case when there is no lamp or the lamp fails to ignite.

Electronic ballasts for fluorescents typically use a circuit which monitors high resonant voltage for the time lamps string is gong to ignite. Any longer than a fraction of a second, circuit goes to shutdown. Too high voltage over the tubes in the running phase also triggers a shutdown. Thus a stress on the ballast components is limited, and also cracking of the tube with EOL electrodes that have spent all of the activation may be prevented. Many electronic ballast for fluorescents also have a circuit to monitor filament continuity. But adding such a circuit is complicated, and only a limited number of filaments, not all of them, is monitored. So, this circuit works more for 'lamp present check' and 'lamp change reset' functionality.

Electronic ballasts for HIDs today most universally employ a microcontroller, so may be made quite clever. It is programmed to catch abnormal lamp behavior like voltage too high, power too high, voltage too low and failure to run up, failure to ignite and so on. Some implementations are even attempting to detect a dreaded dangerous EOL behavior with arcing in the outer bulb or over the base.