If you install the version 2.x of the Date module, you will be able to use more form fields, including a "date_popup" field, which shows a pop-up to select the date.
The module has an option that allows the user to decide if the pop-up should be shown; if you want to override that option, you should use
variable_set('date_popup_timepicker', 'default'). That is the default option used by the module, but if the user selects "none," then the pop-up will not appear.
About setting a custom format with the "date_popup" field, the form field uses the "#date_format" attribute that, in the development snapshot 7.x-2.x, is said to accept most of the formats. The default value for that attribute is the one returned by
variable_get('date_format_short', 'm/d/Y - H:i').
The complete list of the attributes used by the form field, reported in the comment for
date_popup_element_info(), is the following one:
The local timezone to be used to create this date.
Unlike earlier versions of this popup, most formats will work.
Increment minutes and seconds by this amount, default is 1.
The number of years to go back and forward in a year selector, default is -3:+3 (3 back and 3 forward).
Drupal core modules have only the "date" form field, and it doesn't use any "#widget" attribute. Third-party modules could define new attributes, though; as far as I know, no module defines "#widget" as attribute for form fields.