1. It's generally safe to walk around alone at night. Naif Square is a bit dodgy. Women walking alone late at night are sometimes assumed to be prostitutes which could lead to arrest (unlikely if you're obviously a westerner), or ending up richer by the time you get home.
Sometimes people get picked up for being drunk in public (which is a crime) but this is rare unless you draw attention to yourself like try and borrow a sheikh's car (some US marines did this a week or two ago).
Overall, Dubai is safer than almost any other large city that I can think of.
Don't understand why a dog being black means you have restricted exercise times.
2. Technically yes in family areas, in reality no.
3. Some people survive without a car but I couldn't do it.
4. Technically, you'd be breaking the law that says you can't be in a room or car with a member of the opposite gender you're unrelated to. But in reality, there's rarely a problem (search for some of the topics about couples living together to get more opinions). Tourism would probably be cut in half if that law was strictly followed.