I have several points of interest to raise, so in advance, I apologize for the scattershot nature of this question! Feel free to confine your responses to individual regions or to discrete aspects of this broad inquiry :) :
(a) What was the nature of gladiatorial combat in Rome? When and why did it die out in that culture?
(b) To what extent did regions/cultured influenced by Rome (either through subjugation or cultural exchange through trade/proximity) take to this tradition? Would we, for example, see evidence of this throughout the Roman world, or was it primarily isolated to the Italian peninsula?
(c) Was it common to force prisoners (or slaves, or what have you) to fight for onlookers amusement in such an organized and large-scale fashion? Did we see such things in other empires of the time through East Asia, or the Indian subcontinent? In the Americas?
(b) Like, is bullfighting in Spain a holdover from Roman games?
There's no scholarly consensus for the origin of gladiatorial games (munera). The older tradition had it coming from the Eturscans but a more recent origin has been proposed that it came from the Osco-Samnites to the south. Regardless it carried some kind of association with funerary rituals. But even the ancient Romans were unsure where it came from.
The first recorded instance of gladiators is in 264 BCE of three pairs (prisoners of war) and by 216 BCE it was 22 pairs over three days. The numbers continued to skyrocket as it was used by the nobiles to win favour from voters. Ostensibly it still had a funeral connection, in theory, but Julius Caesar held games in honour of his deceased daughter years after her death - it had become subordinated to politics. Under the empire, restrictions were put in place on who could hold munera and how many gladiators they could exhibit.
The belief is that early fights ended with the death of one of the fighters. This changed as munera ramped up and became a business. Most scholars now think there was a death rate of 5-10% - gladiators were an investment as they needed to be fed and trained specifically to fight in the arena.
Munera was popular across the Roman world - the prevalence of ampitheatres demonstrate it. An older view thought the Greek east was too 'civilized' for this but that's been debunked. 22 ampitheatres have been identified in the east and where they didn't have one they simply used other structures (e.g. theatres).
As for its decline, I believe that's due to Christianity who found the practice distasteful (thanks Christians, now all we're left with is WWE!)