Do you have a 12V power supply? Are you using a driver like a ULN2003 between motor and Arduino?https://www.elecrow.com/uln2003-stepper-motor-driver-p-262.htmlIf I recall correctly the 5V 28BYJ-28 draws around 120mA, so 5 * .12 = 0.6 Watts input and about 0.4 ~ 0.5W mechanical output (?). How heavy is your coop door and how fast to raise it?
Running motors from the 5V pin is not good, could burn out the 5V regulator if running from batteries or damage your computer's USB if running from that. 12V motor needs 12V (at least 1 Amp) supply and a driver like ULN2003 to handle the current, don't try to run it from the output pins, they cannot supply enough current, also driver boards have spike suppressing diodes built in. I've gotten about 13 RPM before it starts missing steps but YMMV. If you're planning to use a hoist arrangement, what will your windlass drum diameter be?
I have only the 5V version of 28BYJ-48.I read somewhere that even that version can be powered with 12V.Since you said powering your 12V stepper with 5V is possible, it is likely to turn a bit slower than powerded with 12V.In general I read that it is dangerous to drive the stepper from Arduino 5V rail.I have several Arduino Unos and take the risk until really something breaks.I do drive 2 such steppers from single Uno 5V and have not seen any issues sofar.First I did control the stepper PT camera system from Arduino.Lately I changed to Raspberry Pi (for real time automatic camera centering).I did not want to risk my PIs, so I still use the Uno, but just as 5V power supply.The bottom stepper has to turn top stepper and 70mm lens, both together 55.5g weight: https://forum.arduino.cc/index.php?topic=647703.msg4369656#msg4369656
The 5V 28BYJ-28 (50Ω coils) draws about 100 ~ 110mA so I would wild guess the 12V model will draw about 5 / 12 * 100 = 42mA.Also, I wouldn't try to run the door very fast, remember, there are very few chickens who are Mensa candidates.
Stepper motors pull full current continuously. They are not usually chosen for low power applications.
42mA is the estimated current for a 12V 28BYJ-48, with a 400g (3.94N) door and pulley radius of .75cm, required lifting torque will be about 2.943 Newton cm or 0.0293 Nm. If you roll on more layers of cable, torque will go higher with each layer (greater diameter), if you wind on 4.5 cm torque will be about 8.83 Ncm. I have no idea of the 28BYJ-48's torque. EDIT: You may be able to turn motor power OFF at the top IF the motor's gear reducer has enough drag to keep the door from sneaking down.