Last name first?

Are there any ways to change the order of family and given name in the name expression according to the language?

Based on the screenshot provided, it appears that the rendering is being accomplished via the "Table" style as set at admin/structure/bibcite/reference/settings. If you change the "Reference page view mode" to "Citation," you can use any of the standard citation formats for displaying the bibliographic information. These formats are much more robust, and most formats (such as APA style) automatically display the last name first.

My understanding is that the public display of references really should be using "Citation" rather than "Table" display, where the "Table" display is really only intended for internal use to list all of the data associated with a reference.

Of course, you may really want to use the "Table" display. Since this is a Drupal view mode for an entity type, you should be able to use the "Manage Display" interface for "Contributors" at /admin/structure/bibcite/contributor/settings/display to rearrange that display.