@Zheo

"No, the Horde DOES love Garrosh....so much, in fact, we offer the guided tour right to his throne room for any and all Alliance raids."

As a soldier you protect the position and not the person on it. Every horde should protect their Warchief, whoever's on the throne at that moment. Settle your personal vendetta at your personal convenience, if an enemy attacks you protect your "Throne".

@Scorchhellfire, @Snuzzle

Horde loves (loved?) Garrosh because he was a "hotheaded, act before think after" kinda guy. Everyone loves such a guy, don't they? Just like how you love Cairne right now. His actions were rash, irrational, unreasoned and without any base of authenticity. He acted. Just like Garrosh used to.
Honor combat, apart from the physical combat, also involves the act of "staring down" your opponent. And I believe that's what Garrosh was trying to achieve.
"Garrosh can't believe it. He makes a gambit"
"Cairne instantly agrees. Again, Garrosh can't believe it."
It takes a lot of courage to say NO. Garrosh was never that brave. I am not justifying Garrosh's decision of accepting the duel, but that's generally how one would respond if they were challenged.