When choosing a date range to filter results by on the front-end, all the month names appear to be hard-coded in English and the calendar layout is American (Sunday to Saturday) instead of Monday to Sunday as in the rest of the world. 

Please localize these strings.

Thanks, Milan.  I was able to localize it like that.

What a messy thread, though!

You would do your users a great service by localizing the date picker in accordance with the setting under WPDT > Settings.

Actually, I think ideally, you would provide, in addition to the existing language choices in the dropdown under settings, two additional choices:

1) match site language ( get_locale() ) and

2) match user language ( get_user_locale() )

1 should be the default setting in my opinion. 

get_user_locale() already falls back to get_locale for logged out users, which is very handy!

This would make it so users only have to change the main site language or a test user's setting in order to affect localization across all of WordPress.