Of course yes! Tournaments were a classic feature in every medieval-like settings, so they should be (even not as mastodontic as an arena would make them) in every game with such setting. BTW, A nordish arena would be very great, wouldn't it?

Yeah. The arena was a bit underwhelming, but a great concept that I could definitely see being great if it was given a bit more tlc.

Yes but rather than having you fight other humanoids from the beginning, you would be put to test in front of the audience: fighting against creatures and such. Depending on how fast you beat them and your personality/strength/endurance, the crowd would want you to continue. But camping with a bow on one of the pillars on the map would lead to a drastically lower fan factor, making you return on another weak to try again or after you have pumped your personality/strength/endurance.
Just an idea, though, i'm not sure how it would work practically but it would be cool.

Also, I don't think the combatants would be out right killed, once you down the opponent you'll have a "karma" moment, spare his life and let him walk away from the arena, his hopes crushed for further seeking of glory and utterly humiliated, or finish him of right there on the spot? This could lead to allies, enemies, partners and assassins comming after you. Depending on your choice.

Factor in personality ? I like that a lot. It works completely in the context, too. Yes, more personality and crowd interaction would make the arena experience better. Maybe also add a "fan" problem ? I mean not the annoying one, but let's say that outside, after a winning streak, maybe you should walk carefully in case fans of the other teams would try to get back at you for their betting losses, or try to even out the odds for later fights.

Arena would be cool

In your signature, are you really just getting Morrowind for the 1st time, NOW? I hope you like it. It is probably the greatest game ever created.