Chicken coop door opener circuit
Honestly this day in age, I would just use an arduino. I would also replace that BJT with a mosfet, or just use an arduino-ready relay board.

If you can find some kind of power bank with a built in solar panel that would take care of the solar and charge controller in one shot. 

Those door actuators can draw 2 amps or more at 12v so make sure your supply can handle that. I find that adding a 2ohm series resistor often helps to limit the max current.

Also, I only googled it briefly, but I would not connect your solar panel to battery like that, it could overcharge the battery. I think you need a charge controller, or at least some kind of regulator.

There's a goofy low-tech approach you could consider. Just hook the solar panel straight to a relay coil. Put the battery and actuator on the NO contacts of the relay. That way when the sun gets bright enough, the relay contacts will close and the actuator will fire. To keep the actuator from staying on and draining the battery, just put a switch in series that opens when the door opens.

Edit for update 2:

That actuator will want to draw 1A or more. At 12v it will draw more than 2A. You'll need a bigger battery, like a lead acid or 2 lantern batteries in series.

I would wire it like this:







simulate this circuit &ndash; Schematic created using CircuitLab

Another way you could do it, which is a little more complicated but which may switch on the relay more cleanly is this:







simulate this circuit

Basically it provides a path for the lantern batteries to power the relay briefly. I requires a couple of extra diodes.