Ability to have a calendar on team-managed projects

To align and plan multiple levels of work and private life, to see deadlines. To get an understanding for when I have to/can do that work. To be proactive rather than reactive. To have control and to be productive.
This is what you should do: create a project calendar feed URL containing start and/or due date issues.
I think that there is no need for a fancy full synchronisation feature where you can edit dates from the calendar application. In other words: all changes happen in JIRA. But relevant dates can be seen in the calendar.