What made gladiator fights go out of style in the Roman Empire and its former territories? Did gladiator fights have any influence on the development of bull-fighting in Iberia? Did gladiator fights influence the development of jousting tourneys in France and Germany?
If Roman gladiator fights lasted after the end of the Western empire, how did the various successor states treat combat for sport? Do we know what the Germans, the Byzantine Greeks, and the Umayyad/Rashidun Arabs thought of such fights?
Violent fights arranged between people, watched by an audience for entertainment, existed before the first Roman gladiatorial game was held in 264 BC. The official games began in 105 BC.When we think of a "Roman gladiator", we already have defined him as both Roman and as a gladiator. The last such game was held in 404 AD.
Similar fights did survive the conversion, but not under the same name or traditions. You no longer had official distinctions and terms such as provocatores for fighters clad in breastplate or laquerarii for combatants wielding only lassos. You would rather see a dirty brawl: a cockfight with people. Think modern-day underground MMA fights like they are portrayed in movies, but with deadly weapons. And without the pomp and glory that some lucky gladiators might enjoy.
A good amount of gladiators started their lives in the Roman empire as prisoners of war, or born into slavery. Any person returning to their homelands after becoming a freeman may well bring memories of the arena, or even direct experience in gladiatorial fighting.
As for your question about bullfighting, one class of gladiator was known as the bestiarii, and their specialty was to fight/wrestle tigers, lions and leopards. While I couldn't find anything to support this, it is not a great leap from the Colosseum to a bullfighting arena, or from an angry lion to a toro loco. Keep in mind that this is pure speculation from my side.
(Barchiesi, A. The Oxford Handbook of Roman Studies. Oxford University Press, USA, 2010.
Grant, M. The History of Rome. Faber & Faber, London, 1993.)
It's worth noting that one of the main reasons gladiatorial fights went out of fashion was their incredible cost. We have two partial copies of a very interesting decree of the senate from the reign of Marcus Aurelius (mid-second century AD) which makes it clear how much trouble provincial elites had in buying gladiators and putting on shows without imperial funding and patronage. In this sense, the death blow for gladiatorial games was not Christianity, but the collapse of central administration in the third century, making it more and more difficult to fund games throughout the empire and giving local aristocracies less incentive to do so. Although gladiatorial games would continue to be held, they never regained their cultural prominence. Chariot races, which were already very popular, became the dominant form of popular entertainment in the Roman Empire.