Can I use essential oils to help with pain and sleep?

Yes. But they are unlikely to help especially if you have an organic cause of pain. Aromatherapy helps with relaxation which helps with minor pains and sleep problems. Chronic pain will affect sleep. Talk to a pain specialist.
YES and NO. There are many studies about essential oils and their benefits. But you have to be careful with the type that you may want to use. The oils are not regulated like medications by the fda, so not all the oils are the same. I would talk to a certified holistic practicioner.