The Ouroboros, Lyra Bastion

Question : Do I get to choose the card I put in Reserve with the ability of the Axiom version of Ouroboros, Lyra Bastion?

Answer : Yes. First you play the card, then its abilities activate, so the other effect is already active when the effect takes place.

Question : Does the replacement effect of the Axiom version of "Ouroboros, Lyra Bastion" count as Resupply for effects such as Quetzalcoatl?

Answer : No. Note that it doesn't count as drawing cards either.

Question : If I control the Rare, Axiom version of "The Ouroboros, Lyra Bastion" and I do an Exhausted Resupply, what happens?

Answer : An Exhausted Resupply is a (special case of) Resupply, and will get fully replaced by the effect of the Bastion. You'll therefore reveal two cards, put one in Reserve, and won't exhaust it.