Can rotator cuff injury cause back stiffness/soreness? Also after the pain subsides would foam rolling help faster healing along with exercise?

It can. Usually more so around the upper back by the scapula(the floating bone in the upper back) as this is a place shoulder pain can "refer" to. Foam rolling won't necessarily help with a rotator cuff injury in particular but may help reduce some of the increased tone in the surrounding muscles of the upper back, posterior shoulder that may be affected from a shoulder/rotator cuff injury. Best of luck.