Wasting a whole air cylinder for the deadbolt does strike me as a bit excessive, but if that's what you've got, then why not? Its certainly better to have a well proven mechanism, and 16 months of successful operation qualifies. I would recommend adding some sort of 'trapped chicken' sensor on the edge of the door or doorway that stops and reverses the door if there's any obstruction while its closing before it reaches the fully closed position. A dowel mounted between the levers of two lever arm microswitches such that the vertical edge of the door just clears it when closing might be suitable, though as the door has a raised threshold, you should probably put a sensor for the bottom edge as well.
As it isn't an outbuilding there's no point in trying to economise on power consumption past what's simple and easy. There's mains power available not too far away so simply float a SLA using a traditional linear supply for long life and reliability to power it all and concentrate on improving the algorithm for the door open/close times and the UI for adjusting it. Any electrical system with stored energy is a potential fire risk, but a suitably fused low voltage one is a minimal risk due to the low energies involved and the greatly reduced risk of insulation breakdown.
On the UI side of things, why not move the Arduino down to a reasonably accessible position and fit it with a RTC module, a LCD shield and some buttons to emulate the operation of the mains timer that your users are already familiar with?
None of the chicken sensor ideas appeal to me apart from individual RFID leg rings. That + two readers inside and outside of the door just far enough apart that they don't interfere with each other, so for any specific bird, if the inside sensor was the last to see the tag, you know that bird is in, and similarly for the outside sensor and out. However, as I mentioned earlier, doing anything except raising an alarm if one is missing at closing time is counter-productive. You still need to close the door to safeguard the other birds, and certainly don't want to open the door again during the night for a fox with the tag in its stomach.