My father owns a farm we have cattle guards on the roads to keep them in and we can drive in and out. He has pigs, cow, chickens, ginnies.
This is what I would do. Timers will work you to death keep you busy resetting them as the days get longer and longer then shorter and shorter as the seasons change so I would not use times.
You need a sun activited switch photo cell. They are available at any industrial electrical supply store in most towns. They cost about $20. It is the same thing that turns on city street lights when the sun goes down and turns them off when the sun comes up. It has a sensitity adjustment.
I have noticed that our chickens go in the chicken house on their own when the sun goes down. It is not completely dark yet and the chickens are all in the chicken house already. So I think a servo motor and a photo switch will do the job. You can also buy louver vent servo motors at any industrial electrical supply store they are used on furnaces and air vents and boilers to open and close or move something. Attach the servo to the door so when the sun goes down the door closes, when the sun comes up the door opens. That should work fine. If your chickens are not all inside the house when the door closes then you may want to put a delay timer relay on the servo motor so when the photo electric cell tells the motor to close the door there is a time delay. Potter & Brunfield sells adjustable timer relays get one with a 20 minute delay and you can turn the dial to 10 minutes or what every is needed. You will not need a timer in the morning.
Buy a 12 VDC to 120 VAC inverter to converter your 12 volt power supply to 120 volts. You can buy a 300 watt unit for $30 on ebay or any of those foreign tool stores like Northern Tool. The photo cell has 3 wires and is very low power. The servo motor is low power too only about 1 amp. Servo motor comes with an adjustable arm so you can set it for any stroke or lenght you need.
McMaster Carr and WW Grainger both sell what you need and you can order online.
I have been searching both MMC and WWG for the servo motor but so far I have not found it. I know they sell them because I have orders them before. when I find the servos, relays, photo cells I will post a link here.
Photo switch $10
https://www.grainger.com/Grainger/items/6P007?BaseItem=1JLG4
Timer relay. This is the type relay you need but you need. Pick one with about 15 to 20 minute delay. **broken link removed**
I can not find a pic of the servo you need. We have several of these at work. We have them on the building vents. When the fan comes on the servo opens the louver vents. We also have these servos on 11 large furnaces. The furnace ignites on low then when the control show the furnace is light the servo opens the gas valve and the air valve both at the same time to power up to 1,000,000. BTUs. These are very simple servos they have a 180 deg rotation with an arm that is about 6" long with several taps to conncet an arm that can open a door. The servos box is about 4" x 4" x 4" square 120 volts AC. I don't know the name and model number so I am not able to find a pic online. The way this servo works is the arm stay retracted all the time with 120 volts on it the motor hits a limit switch built inside and it stops in the netural position. The photo switch triggers it to rotate shen the sun goes down it rotates 180 degrees to close you chicken house door. After it rotates the full 180 degrees it hits another limit switch build inside that stops it there. When the sun comes up the photo switch turn off the trigger to the servo so it returns to neutral position that will open the door the next morning.