What time of the month is it best to have unprotected sex and not be pregnant?

Not a good strategy. Timing is not a very good strategy to prevent pregnancy. There are too many ways it could fail. During the late luteal phase through the start of the menstrual flow fertility is at its lowest, but timing this is tricky, and timing is not a reliable indicator.
There isn't one. Although you are more likely to get pregnant during an ovulation period, a woman can still get pregnant at any time of the month with unprotected sex (sperm can survive for a good period of time in the uterus and fallopian tubes). If you do not want to get pregnant, there are several options for birth control you can choose from.