Toodledo might be worth a look. As task managers go, it's configurable with the usual due date, time, folder, priority, etc. fields. It also enables tracking of short-term, long-term, and life-long goals, to which individual tasks contribute. From the website:
Setting personal goals is a powerful way to direct your energy and determine what you want to achieve in life. Unlike tasks, which have a clear path of action, goals are usually vague and difficult to quantify. Once you define your goals, you can assign them to individual tasks and keep track of your progress. This is a helpful way to see which goals you are progressing on and which goals need more attention.
Regarding some of your specific requirements:
- The website has a calendar view, but it's limited to two months at a time and only seems to display the number of tasks due on a particular day.
- There are various mobile apps available: a first-party client for iOS devices costing $2.99, and various third-party clients for other platforms as well.
- Tasks can be sorted by any of the fields, and you can also show (use) or hide (not use) any of those fields at will. In your example, you could sort by due date, importance, status, and so on.
It might not be a perfect fit for your needs, but free accounts are available if you want to try it out. See also this comparison on the features and restrictions of free vs. paid accounts.