Hi John.
Welcome to the forums! 
Lovely idea. Such a great beginner project! Simple, useful, and personal.
Let’s start with some tools
The above looks promising.
I think I would just improvise a gear and pulley system which connects to the door.
I haven’t personally used this motor personally, but I like that it’s 5v.
That’s good because alot of solar panels, like this one below, are 5v too.
Pick your favourite microcontroller. I don’t think you’ll need wifi so something simple like this below will be fine.
and also
Maybe, but we’d need a model number and a picture to comment. 
I think @Jaryd has you covered below.
Finalise your design, then commit
Consider sketching up your design on paper before commuting to a purchase.
Don’t hesitate to post your sketch here to get a second opinion.
Good luck.
Pix 
