1. Our database store as You wish. The justification depends on the requests. If there is selection on weeks, etc. in date, if not, you can simply int (unix)
2. To convert to a widget before making the base in the photo above, it is possible through Yii::$app->formater, you can connect the component demos.krajee.com/datecontrol
3. DatePicker supports setting a date type.
Total to get started, decide how you will store in the database, then set the type of date in the widget, then if necessary add to the model beforeValidate, afterFind, etc.