Apparently my group has been playing this wrong.

We've been playing that all cards ready during the "Ready" step of the Status phase. Apparently that is incorrect and a hero's Class cards, Items, and Rewards ready at the start of that hero's activation?

Does that mean a hero gets to use cards that exhaust when used twice if they get 2 activations in a turn?

That's correct, they do get to use them twice with 2 activations. Additionally, cards start the mission ready, so abilities like Gideon's "Called Shot" could be used once before and once after his activation in the first round.

And of course, be aware of the difference between "Exhaust" abilities and "Limit once per round/activation token" abilities (not that you asked, but just in case ).