Let's make some crude guesses to paint a very rough picture.
Suppose your motor draws 2 amps for 30 seconds when it is raising or lowering the door, and draws no current when it is not doing anything. Suppose it only lifts and lowers once each per day. That would be 2 amp minutes per day. If it works at 5v then that would be 10 watt minutes per day or 0.17 watt hrs per day
Suppose your Arduino and other stuff connected to it draws 100mA all the time (24/7) also at 5v. That will be 0.5 watts for 24 hrs or 12 watt hrs per day.
Your solar panel will need to be able to provide more than that (say 20 watt hrs per day) on average in the worst winter weather when there is very little sunshine.
A 5 watt panel will only produce that output in brilliant sunshine - so guess maybe 1 watt, or even 0.5 watts on ordinary days. And in winter it might only do that for 4 or 5 hours per day giving maybe 0.5 * 5 = 2.5 watt hrs per day. That would be only 1/10th of what you need.
All these numbers are guesswork just intended to illustrate how to do the calculations. You need to use the proper numbers to get a realistic indication of what is needed. (And you need to check my maths 
)
Also take account of chicken sh*t on the solar panels.
...R