Hello, I work with a rescue and can tell you that you are in for a treat!
We have had issues with the running back and forth etc.
First, I would not recommend walking her EVERY time she runs to the door, that will set a training method (to you, not her--she's training you to do the walks).
The barking is most likely the new environment, noises, or wanting to be with someone.
I saw on here that someone recommended a walkie talkie in the room with the dog and then kept one with them. When the dog would start whimpering they would talk and reassure the dog and after a while it worked well.
I would also recommend that you keep a radio going (not too loud) to provide background noise that she may be hearing as "strange" noises to her.
When changing the food, combine the old and the new gradually, it will help with diarrhea, what was the "old" brand of dog food? Royal Canin is sold here in the states too, I don't know of the origin (may be a British company), you will get LOTS of advice here on dog foods, I don't know what is readily available in the UK, but Arcana and Orijen have been highly recommended by several here and other places.
You may also want to consider investing in some re-usable potty pads in the beginning to help her in case one of the "outside" times is legitimately a potty break.
I know that each time we get new rescue as one is adopted out, it is like first night at home with a baby and just takes time.
Did the adopter provide any type of blanket or anything from the old home that would have the "scent" on it? This can also help to reduce the anxiety with a familiar smell.
Hope this helps some in the interim, keep looking on here, there is definitely A LOT of information here to be had!
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