[Balanor]

Does killing a shackled slave or letting them be killed count against notoriety? I know it only says "escape", but I'd like to be sure because I often do those things to be able to capture a "better" slave.

No. Notoriety is only lost if you try to hold on to the 'unbroken' slave for too long and they escape.

Do fights in arenas increase notoriety more than simply killing the same creature in the wild or in a dungeon? And if not, shouldn't they? :)

Yes it does. By alot compared to what you'd get for the same kill outside the arena. But as with almost everything that awards notoriety, there is a cooldown timer for how often you get notoriety for an arena kill. So while you can spend all day fighting in the arenas, notoriety will only be awarded every so often.