I agree that the sleep issue is definitely the one to tackle. Once your child learns to sleep through the night, a lot of these issues will go away. A 7pm bedtime seems pretty early for that age (unless she has to get up at 6am), and with the naps, I'm not surprised she can't sleep through the night.
I'd suggest that you don't try putting the younger kid to bed after the older one is asleep, because the older one is likely to resent that, and one of the ways she might rebel is to keep herself up later and later.
Inexplicable bouts of irritability is how my daughter announces her lack of sufficient sleep, so they go hand in hand.
