Not sure about the staff or cure disease potion, but as for the armor; there are armors that she just isn't programmed to wear. If you're dedicated I'm sure the nexus has some kind of mod that will allow for her to wear those armors.
Try this mod - http://www.nexusmods.com/skyrim/mods/14037/?
It's the best for everything related to followers, also has a choice to make them wear what you want them to.
Also highly recommend this one - http://www.nexusmods.com/skyrim/mods/18045/?
It's compatible with UFO and comes in very useful for maintaining follower locations when you send them home.
Tristan's post in playing dress-up with Lydia issues was marked as the answer