“Is local anesthesia for hand surgery safe?”

Local anesthesia is absolutely safe for hand surgery.  Depending on the type of surgery you're having you may get local at the site or a nerve block further up the arm or shoulder to completely cover the area of the surgery. You will probably also get some iv anesthesia to keep you relaxed but it is all safe.

Yes, a local anesthetic block is perfectly safe for both anesthesia and analgesia.