Yes, the gender-distincted way is the preferred way to correctly address people in German. May it be
For a group of people the generic masculine is still common, but sometimes frowned upon. Example
Studenten would normally mean all students but people prefer today
Studentinnen und Studenten or even, if possible and existing, the nominalisiertes Partizip I/Adjektiv (nominalized adjective/Partizip I) of a fitting verb:
There's gender confusion here, but not in a previous example. 'The female employer doesn't have a car' is not accepted, even though gender identification was in an earlier example. If the translators don't want to identify the employer (here) as female then use a neutral word. Germans don't have issues with this, only American politically-motivated translators.
That phrasing sounds very awkward. Use "has no car" or "does not have a car."
Also remember that kein applies to Auto. "He has no car" translates to "Er hat kein Auto." No is modifying car, and kein is modifying Auto. When you translate the sentence, you must remember to keep it that way.
The exception is when you say "does not." That is a valid translation, although it is less literal.