Are there bus after midnight at Beijing airport? Can we stay inside the airport lobby at night?

Yes, you can stay inside the airport, but it may be very cold at night in the airport. It won't be comfortable. Thus you can take the Beijing Capital Airport Bus(https://www.travelchinaguide.com/cityguides/beijing/transportation/airport-shuttle.htm) line 4 from airport to International Center, then catch a taxi to Dong Si Area.