Is it possible to create a drop-down calendar in a form so that the end used can select a date rather than type one in?

In 10 easy steps!  

Creating a drop down calendar in Adobe Pro:  1) Enable the form.  2) Click Prepare form 3) Right click the and select properties on the field you want to make a drop calendar from.  4) On the general tab name the field Date (you can name what you want and on tool tip I write in "Drop Down Calendar).  5) Appearance tab, format to your liking  6) On Position tab, format to your liking (you usually already have this formated in line with your form)  7)  Options tab, choose your alignment and check only "Scroll long text"

😎  Actions tab, on "Select Trigger:"  Choose "Mouse Up"  and on "Select Action:" choose "Run JavaScript"  Next click on the "Run a JavaScript" that you just chose and enter the following script:  FormRouter_PlaceCalendar(this.getField("DATE"),false,"dd-mmm-yyyy");  9)  On the format tab on the "Select format category:  Select "Date"  and under the Date Options MAKE SURE THE DATE FORMAT MATCHES THE DATE FORMAT YOU HAVE IN THE SCRIPT.    10)  Close it out and select the preview mode on your form to see if it works (it should).  

In the latest version of Adobe Acrobat DC use the calendar icon.