How much of your two weeks is likely to be taken up with work? I wouldn't suggest allowing less than about a week to visit any of these cities - you could easily spend a month in and around Sydney and not have exhausted it. Maybe you should think about only visiting one of them, and making it the base for side trips to Melbourne and Auckland for business.

A standard GSM phone (anything that works in Europe/Asia etc.) will be fine for both Australia and NZ.

Sydney city beaches are within easy reach of the city centre, so you can just plan it into a day's activities. I would recommend avoiding Bondi Beach, though. It's always seemed too crowded and busy at weekends. I have had pleasant times at Palm Beach, and was taken to Warriewood Beach a couple of weeks ago - it has few facilities and is relatively uncrowded but you can drive right up to the beach and park there.

If you want to get away from the city altogether for a beach weekend, see if you can get accommodation near Gerroa. It's about 2 hours drive south of Sydney, beyond Wollongong and just beyond Kiama. Not for nothing is the beach called Seven Mile Beach, and it's a very user-friendly beach as well.