1. There is a guideline that families cannot be "parked" in US. So if the L-1 holder is leaving US for good in Jun 17, the expectation is that you should leave at the same time. It would better if you apply for COS before this date.
2. As long as you have COS filed, you can continue to stay and study on L-2. If COS is denied, then you need to leave US ASAP.