The rules and various cards talk about a routed hero (one who is already routed). I'm not seeing anything though that would cause a hero to become routed. Have I missed something in the rules or some specific card(s) that can cause a hero to route?

The rules say that it's very rare or near impossible for the hero to be routed, but they made rules just in case it does happen; that makes me assume that it may be possible to rout a hero via a tactics card or special ability, but not through battle/duels alone

I so far haven't seen any cards or abilities that would rout a hero, so for now I'm assuming that I either haven't seen the card/ability, or that they are planning ahead for maybe an expansion or future errata/rules

so basically, don't even teach that rule and if it ever happens in the game then refer to the section

We had one quest last night (a duel quest) where the hero would have routed had he failed. He succeeded the quest on the final round of the duel!

Theres a couple quests where the hero must retreat if he fails

Yup, my first Quest attempt ended in Mad Carthos being routed.