How can I expose a view with date filter?
In 7 you can do this with Better Exposed Filters by selecting the jQuery UI datepicker option in the screen of your screenshot. There is an alpha out for 8.


  The Better Exposed Filters module replaces the Views' default single-
  or multi-select boxes with radio buttons or checkboxes, respectively.
  Description fields and Select All/None links can be added to exposed
  filters to make for a better user experience.
  
  Views Filters is a powerful tool to refine the results returned by a
  given view. When you expose a filter, you allow the user to interact
  with the view making it easy to build a customized, advanced search.
  For example, exposing the node type field as a filter lets your site
  visitor limit their search queries to just blog entries. Better
  Exposed Filters gives you greater control over the rendering of
  exposed filters.