Is the Nemesis boss spawn chance calculated independently for each kill, or is there some hidden system (e.g. accumulated chance or spawn cycles)?
Many Nemesis bosses have a cooldown window. For example, Dharalion has a cooldown window of 6-8 days. If, for example, he is killed today, that means you should check him again in 6-8 days.
If a boss is not killed within the window period, it means it has poofed*. When this happens, you should keep checking until you find it again (or until someone else kills it). Some bosses spawn with other creatures, and unlike bosses, these creatures do not poof. So, it's worth paying attention to that.
*In Zao, the spawning of some bosses is not guaranteed every cycle. Sometimes only the creatures that accompany them spawn. When this happens, consider it as if the boss had spawned.
For more information: https://www.tibiaqa.com/31520/nemesis-bosses-different-strategies-maximize-bosstiary-points-killing?show=31520#q31520
Does kill speed or number of creatures killed in a short time affect the probability?
No.
Are there any known mechanics related to server save, respawn cycles, or area resets that influence Nemesis spawns?
For the Bank Robbery bosses, it's good to know the time the server comes back online after a server save. If the server comes back online at 10:07, check the World Board in the Adventurer's Guild every hour, xx:07. If the server goes offline for maintenance at any point during the day, this will also affect the time. These bosses are completely random and don't have a known cooldown window.
There are some bosses that have a forced spawn. You can find more information here:
https://www.tibiaqa.com/31304/what-are-the-nemesis-bosses-that-can-be-forced-to-respawn?show=31304#q31304
Is there any difference between hunting slowly (fully clearing spawns) versus rapidly rotating multiple spawns?
No.