I don't think they recharge weapons, they might, but I don't think they do. I always give my follower a soul trap weapon and a bunch of empty soul gems and they actually do run out of charges sometimes and never used the filled ones they had.
I know they use health potions, maybe even magicka, but I haven't seen one use a poison or stamina one yet. I don't think they use fortify health though but I could be wrong.
The first time they used a summon staff was pretty cool. I heard they can use scrolls but I never noticed Lydia using one yet.
Edit: I am pretty sure they eat certain food also, maybe all. I gave Lydia a bunch of apples or something and I am almost positive she ate them when she ran out of health potions.