You cannot retrieve the default language setting from the Client API, you can however retrieve the default language using CommunityCommons action “GetDefaultLanguage” and pass that to your widget instead.
I believe if you use date attributes in a pluggable widget, the formatter property of the attribute will have the appropriate date format for the user’s locale.
Also, while the date format isn’t in the public client API, you can still get to it at:
mx.session.sessionData.locale.patterns
That API is subject to change of course but it has been stable for a little while.