Everything which has been said should be of great help. A few more things.

Hamburg is pretty safe in most parts of town around the clock and the entire city should be safer then any American big city. Especially walking with a dog, nobody would ever attack you Find the apartment in a place that is convenient for your needs which should also consider the daily commute for your husband. Hamburg is not that small, the commute can easily eat up plenty of time.

As far as having a car: in Germany, car sharing companies are rising in each city. In my opinion, one doesn't really need a car in the big cities but if it is more convenient to take the car from time to time, just use regular rental agencies or do the car sharing thing. Companies like Car2go, etc. charge you by the minute and cars can be parked anywhere in the city in a legal spot.

As far as the dog, dog sitters and all the facilities you have in the US for animals are not that common but you will find them. In general, Germany is a very animal friendly country.