I probably have my own ideas to add to this discussion, but I'm just too hyped up on coffee right now to be able to focus on any of them. However, I do have thoughts on some of the ones presented already.
I like the idea of putting Arena level (which I assume includes the level up meter, Rut) inside the Arena page. I hate having to constantly switch back and forth to check on my progress while score grinding in the Arena, so I'll + that suggestion.
Quest button being removed would be good. If there's a plan for future daily quests that might be a problem though. Upgrade menu in the bazaar makes far more sense imo.
I'd keep the level line as well as the type line. Most people refer to each of the opponents as AI3 and are usually understood when doing so because it's stated on the level line. Otherwise people may get confused much more easily as I doubt it would make sense to refer to AI0 as AI0 after that line's removed.