A) How did slaves that were to become gladiators work within House Redoran? Did they fight in the Vivec Arena?
I can't recall anything about Redoran gladiators (although it's been a while since I've played Morrowind) but if there are gladiators, then yes, they'd probably fight at the Arena. Unless there happens to be a mainland Morrowind arena that I don't know about.
B) Was it like in real life with the Roman Republic and the Roman Empire? Such as ludi(gladiator school), and so on so forth?
I never heard of one, but I'd suspect you're expected to train yourself or seek training on your own time. I could be wrong, but I never heard of a gladiator school in TES.
C) Cyrodiil did not have slavery in the Third Era, so did that mean slaves from provinces such as Morrowind were permitted to fight in the Imperial City arena?
It would depend on their laws. If their laws said that a slave in Morrowind remained a slave in Cyrodiil, than yes, they could probably be made to fight. If slavery was simply not allowed at all in Cyrodiil regardless of the legality of where the slave was brought from, then probably not. But even so, anyone can choose to fight in the Arena. And it's amazing how so many people do, considering how every match is to the death.
If there is any good information regarding slaves and gladiators in the Elder Scrolls please do tell me. I know many of you may have some lore to share
Other than that, the only arenas I know of are the Kvatch Arena (which was destroyed), the Cann Arena (which is also in ruins and while combat apparently took place there around the time of TES IV, not only did the Heretics not intend it, but only one of the participants was a gladiator), the Rotheran Arena (which while it's still intact, again, most of the participants are unwilling slaves and prisoners with no combat training), and the Felgourad Arena (which is only http://www.uesp.net/wiki/Shivering:Scented_Parchment by a supposed gladiator champion in Cann, and I've never heard of such a place in Tamriel)