Translation:The little girl has finally gotten a dog and is very happy.
In english I would de tempted to add a "she": ...and she is very... . Can that be done in Danish?
Yes it can but it is not neccesary as it is implied that it is still hers. Just like in English.
Gotten is not a word in English. I think it is a part of the American language.
Gotten is indeed a word in English. Americans speak English, so it's allowed. You may not like it for some reason, but your feelings aside, it's valid.
Duolingo defaults to American English, as unfortunate a choice as that is.