Can we stay in the airport overnight to catch a flight in the early hours of the following morning?

I believe the close the doors during none flights hours specially at night

They don't ticket for overnight, or any length of time, parking. Terminal is closed as described after the last flight arrives until 4 a.m.

The airport terminal is closed and locked overnight from 12am-4am, however the vestibule areas between the sets of doors are open overnight.

They will ticket you for overnight parking. Your best bet would be to stay home if you live here or in a hotel if you don't. Catch a Uber in the morning.