Yes. Performing these procedures exclusively under local is probably the safest way to do them.
They usually do. The anesthesiologist will use local anesthesia to decreased the discomfort associated with the placement of an epidural. Then if the epidural is used for acute pain an infusion of local anesthesia and narcotic is often used to treat pain. In the pain clinics a local anesthetic is not used in the epidural injection, because it the needle is removed and a continuous infusion cannot be used.