Did people root for specific gladiators, types, schools or whatever? I've heard chariot races had intense fandom that was bigger than gladiator matches. Is that true?
Not exactly Rome, but in very early Byzantium, racing fans where hugely influential in society. By the time of Justinian I, there were two big fanbases: the Greens and the Blues, based on the color of the racers' shirts. There used to be Reds and Whites as well, but they had been absorbed into the other two when the Western Empire fell. However, these where not just sports fans. They had a opinion on, and a say in, many aspects of culture and politics, such as religion, which also divided the two factions. They would often riot and fight, causing literally thousands of deaths. These riots ultimately culminated in the Nika riots, where the two factions united to try and overthrow the Emperor. These riots where huge, costing tens of thousands of lives, and broke the power of the Blues and Greens in street politics, though they continued on in imperial ceremonial roles.
Further reading:
Alan Cameron, Circus factions: Blues and Greens at Rome and Byzantium, Clarendon Press, 1976.
An important event in establishing the separation of church and state was triggered by chariot fandom. In 390 at Thessalonica a famous chariot racer was accused of attempting to rape a man and was put in prison--this occurred right before a big race, and the people in Thessalonica demanded his release. The commander of the city was a Goth, and he refused to release the racer.
In this time period the culturally-Greek Roman citizens in Thessalonica wouldn't have cared particularly about a homosexual offense, but it was considered much worse in Gothic culture. The combination of enraged chariot fandom and perceived attempts to impose Gothic values on Roman citizens led the people to revolt against the Gothic commander.
In response to this, the Eastern Emperor Theodosius sent troops to Thessalonica who led a massacre against its citizens. The massacre was so bad that Ambrose, the Bishop of Milan, essentially excommunicated the Emperor (although I'm not sure if they called it that back then---at any rate, he refused to celebrate mass with the Emperor). Theodosius wasn't forgiven until he publicly supplicated himself in a sackcloth before Ambrose. This was one of the first times the church had ever asserted this sort of authority over a secular ruler, and it all started due to a classic sporting-figure sex scandal.