I note that in each of the Armor skills in the Conjuration tree, it says that only one armor skill can be active at a time. Does Diamond Skin count as an Armor spell? Or can I trigger both Diamond Skin and, say Energy Armor or Ice Armor, at the same time?
Diamond Skin is classified as an absorption effect, so you can use an armour effect like Storm Armour along with Diamond Skin with no problems at all.
Yes you can trigger Diamond Skin and "X" Armor at the same time.