Is it possible to enter the park with your 4x4 during hours when the park entrance isn’t manned by a ranger? If I arrive at 4am, will I be able to enter?

Yes you just have to stop at the ranger station and get the gate code and leave money in a drop box

Only if you know the code to the gate. Otherwise you'll have to wait

What happens if I arrive before/after the kiosk is open/closed? There is an automatic pay machine located in the Lower Ranch at the Ranger Station. So I believe you can come any time and you enter the park if you know the code to the gate. Park is opened 24 hours.

If you are camping, there is a Kiosk you can use to "check in" at (under FAQs) http://ohv.parks.ca.gov/?page_id=25986#collapse2Four

No