When a back injury starts healing is it normal to feel pain elsewhere?

Not usually however. Sometimes you can injure a disc in your back and as the back pain improves the disc is actually herniating & compresses a nerve causing radiating pain into a leg(s). It could also be that the back injury so restricted you that when you resume more activities you are deconditioned and feel muscular pains in those areas. If they persist or worsen, see your physician.