Can tennis elbow cause numbness in fingers and pain shooting not only down the forearm but to shoulder too?

Certainly can... As it causes local inflammation even though "tennis elbow" isn't really an inflammatory process. It can irritate the subcutaneous nerves and you compensate with your shoulder and wrist and they become sore. Treatment typically with wrist brace to rest the tendons, ice massage, and ideally a cortisone injection with pt. Nsaids don't work because it is a necrotic phenomenon not inflammatory.