My mother was there, but she had better things to do than kick a soccer ball around. So she said, anyway. At the time I didn't think there was anything better than kicking a soccer ball around, especially not the likes of laundry or dishes or vacuuming, but my mother didn't agree. And the danger of being home alone with her was that she'd recruit me to help her wash or dust or vacuum, and she wouldn't tolerate the dribbling of a soccer ball around the house as I moved from chore to chore.

To play it safe, I waited outside for weeks, just in case the new neighbors moved in early.

Here to play it safe means to play the soccer in a safe way or it has other meaning?
To play something safe means to handle that situation in a way that will (hopefully!) turn out to your advantage.

"The cars were moving very fast so I played it safe and crossed at the traffic lights."
in your example sentence, it refered to "I crossed at the traffic lights", instead of car?
Students: We have free audio pronunciation exercises.
Yes, I am the person doing the crossing.