Machine not answering. new battery. lights on with 0 messages.

When the power went out it may have set the machine to its default setting, "answer function off". Press the "On/Off" button to check for sure. A battery really wouldn't help in this matter because it's only there for backup; when the power goes out the batter will save your settings, outgoing and incoming messages. Perhaps when the power went out you lost your outgoing message, and now, when people call, they don't realize that that they are supposed to leave a message. One more thing you can check is the switch on the side of the phone which tells the machine to answer after 2 or 4 rings.