Hello Nyght,
Thank you for contacting Microsoft Community.
I appreciate your patience in this regard.
Quiet hours will turn on automatically at the times you've set. But you can still turn it on manually. When you go to Quiet hours setting it will give many flexible options that allows you to customize Quiet hours accordingly. Sat for example, you can set date on which the phone should enable the quite hours along with Start and End time. Basis to which Quiet hours will turn on automatically.
It also allows you to Edit break-through list. Where you can allow the emergency calls from desired contacts. Moreover Cortana will also send text to contacts who aren't allowed through, saying that I'm busy at the moment.
With regard to the desired task or event sync from the Calendar app, I suggest you to post your feedback using Windows feedback app, which will helps us to look into this with future updates. This will helps us to improve your experience.
Keep us posted if you require further assistance.