I strongly recommend El Kapallaq for seafood.Its located in Av. Pettit Thouars 4844 in Miraflores and the owners name is Lucho and his wife name is Denise. They will serve you the best food and will treat you very well.

Take a radio taxi to Miraflores to the shopping center on the ocean in front of the Marriott Hotel. There are alot restaurants,bars,cinemas,shops,bowling alley,etc . It's a very safe area and open very late. Also, the entire area around the shopping center is safe and there are many more bars and clubs you can walk to.

Lima Airport Locker

There is an ATM in the arrival area where you can opt for US$ or Soles.
The lockers are pretty big. We put two 22" suitcases & one small bag into one locker. It seems very safe, you get a receipt and the key. Rate is 28 soles per locker per day, less if you pay by the hour.
Can't help with the cell phone question, we never made any international call.
For calls within Peru, buy the "147" call card (I think the cheapest is US$5) that you can use with absolutely every type of phone in Peru. It's very convenient. We bought it at our B&B.