Hi,
My bid is for a system (very simple design - bootstrap), users login with twitter, add/edit/delete tweets, like you describe.
Export to json, xml is also easily done.
A SIMPLE API can also be done. actually the api will be done first and the site will use it's own API.
User details will be limited to what tweeter sends, and there will be no user settings (except to show the API key), no mass upload of tweets, just what you described above.