You can, and I think some people do, but generally TaskPaper is intended to be used by an individual.
TaskPaper is very simple. It’s just a plain text file. If you want to share with others you can store that file on a shared drive like iCloud or Dropbox.
This can work, but you also might sometimes get conflicts if you both edit at the file at the same time. Then you will have to resolve the conflict using maybe a text diff tool. Unless that all sounds simple you might be better with an online todo list that syncs automatically without conflicts.
It is unclear if the dates set up show on the note? or on ical? or maybe both
TaskPaper dates don’t automatically sync with iCal. Instead there are commands to import items from iCal, or export items to iCal. You can find more about dates in TaskPaper here: