Although I don't speak Danish, I was able to figure out what these words could mean—by means of some dictionaries, of course.
The word "mon" simply means "I wonder" in this case, for instance: "Mon han lever endnu." (= I wonder if he is still alive/if he still lives.) I also found that you could translate "Ja mon ikke" as "I suppose so".
The word "dog" is not easy to describe, if I understand it correctly. It may correspond to the German word "doch" and is just a filler or intensifier in a sentence. There is no equivalent word in English, so I'd circumscribe it with "do" or "yet": "Luk dog den dør!" --- "Do shut the door, will you?"