Are you able to push the LEGO city train around the tracks without damaging it or do you always have to use the batteries to make it go?
You can definitely use the trains without batteries and push them on the tracks. In fact, a lot of LEGO trains come without the battery option, and you can add it if you want.
Most people will get the battery option as it is fun to watch LEGO trains go on their own, but it is not necessary.
You can even build raised tracks and have the train go down on them by gravity...(c:
One thing I would also add is that it is best to either pull the train from the front, or push it from behind. Don't put pressure on it from the top; like how little kids tend to push trains putting their weight on the train as they push it from the top. That could damage the axles of the train.