Allow users to sort stories by date or content type
i think you might be confusedly mixing a few different concepts? 

"story" is one of the default/example content types distributed with Drupal and is nothing special, ie. not quite sure what you mean by "story view".

I assume from the way that you're phrasing your question that you're new to Drupal, have you discovered Views and CCK?

exposing filters is relatively easy in Views 2 and 3, but exposing sort options in general is more difficult in Views 2 (you'll probably need some basic PHP knowledge to achieve this) but can be done through the UI in Views 3.

popup calendar widgets for date fields exist in the Date module once you install the jQuery UI module.

if you want a calendar widget that is "inline" (doesn't popup from a dropdown menu) then you'll likely have to do a little coding, but it isn't very hard as jquery UI provides a good API for inserting calendars into your page.