1) "Oder?" added to declarative clauses
It is relatively common in German everyday communication to add oder? to a sentence that is meant as a question in order to signalize explicitly that this is a question and that an answer is expected.
Heute ist schönes Wetter, oder?
Du hast eine neue Hose, oder?
Die Amerikaner waren auf dem Mond, oder?
Die Erde ist eine Scheibe, oder?
Der Wein ist klasse, oder?
Ich habe eine schicke Frisur, oder?
As you see, the oder is to be attached to declarative clauses (Heute ist schönes Wetter), not interrogative clauses (Ist heute schönes Wetter?).
As for regional peculiarities: it is possible that one finds this habit more common in the South of the German speaking (part of the) world. Particularly in Switzerland, it is something like a "national" quirk to add oddrr? to every second sentence, like the British would add isn't it?. This is then less meant to provoke an answer from the collocutor, rather it serves to keep him/her focussed on the communication by testing if he/she still listens and is able to react adequately if needed.
Your example with the girl and her new dog is a bit odd, but it is conceivable. She would use the oder? attachment in order to provoke her interlocutor to an explicit reaction. It sounds extremely expectant, though, even to a level of purposeful exaggeration.
Note that this is common only in oral, informal communication. You wouldn't do this in written communication, would you?
2) "Oder..." added to interrogative clauses
There is also a habit of adding "oder..." to interrogative clauses. This is done especially when the person speaking wants to be friendly, open, and non-demanding:
Haben Sie schon Erfahrung mit Computern, oder...
Sind Sie schon gegen Tetanus geimpft, oder...
If asked without the "oder...", the person asked would be forced to say bluntly "no", which could be perceived as offensive, embarrassing or inappropriate. Therefore a person asking who wants to indicate explicit friendlyness and lack of special expectations can add "oder...", thus indicating that he/she is aware that various scenarios are possible and would be seen as normal and expectable.