I would not want to get the borders of the veneer wet at all really, so a lightly dampened cloth only if you really need to clean a stain. Dry microfiber for dusting.
Indeed. Dry cloth or damp cloth only should be required.
Even most wood veneer speakers have a urethane finish which doesn't require any oils or special cleaners.
I would contact Axiom about how to possibly repair the veneer peel up so it won't get worse if someone picks at it.
If these were B stock speakers when you bought them, you may have just found the B stock flaw.