How do I rewire this circuit controlling several lights and receptacles?
You have an aged system that is seriously compromised.  Based on the question, you seem to have little experience with eletrical work. It is usually better for a novice to begin with straighforward tasks on a healthy system.

I would strongly urge you to bring in a professional. As other have suggested, a full upgrade to the service panel and the wiring is highly desirable. While it may be possible to cure the current problem without a full ovehaul, you need someone very knowledgable with the right equipment to ensure that the repair is sound and safe.