1) If things were removed because they weren't done well previously we wouldn't have a very interesting series. Fix it, don't remove it.
2) Lore-wise they already have an Arena, if I can find a link to the quote I'll post it.
3) Everyday life doesn't end because of war, not even a civil war. Not everyone is a soldier.
That being said if they do put it in I agree with the stuff you said on improving it, the stuff I didn't include in quotes.
1) yep, that's why I had the parenthetical
2) my ignorance
3) ummmm depends on the scope of the war? (ya, grasping at straws). For some reason I picture Skyrim as one big warzone, not really a place with a "front line", rear area, or home front where an Arena type thing could take place.
Pick-up matches between warriors..ya, I can see that. Ohhhh that could even develop into a longer questline than the Grey Prince. Gain a companion, befriend, fight battles alongside..then have to battle in an impromptu Arena...what will you do? Of course, that'd require adding depth of character to the companion...
Now go away...quit making me think!