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