If I draw a Curse that says "Lose Your Armor," and I have an Armor card sideways, can I lose that one to satisfy the Curse?

It depends on the exact wording of the Curse. If the Curse says "Lose the Armor You Are Wearing," it must be Armor that you are using (and therefore gaining the benefits of). If it is "Lose One Armor," you would lose any single Armor card you had in play. And if it is just "Lose Your Armor," discard everything you have in play that is Armor. Cheated Armor is still Armor!
Related Rule(s)