1. There are a lot of family friendly areas. The South Side of HK island (Stanley, Repulse Bay, Tai Tam, Parkview, Shouson Hill, etc.) would be up there with the spots where you find a lot of American families with younger children. A but more green and space, lots of play groups, etc. But, by no means the only area where that is expat friendly. Happy Valley and Pok Fu Lam are also nice places for families with young children. Happy Valley has the advantage of being closest to Central. There are also places in Kowloon and the New Terratories, but I am not as familiar with these areas. Discovery Bay is quite nice. Re: price, on HK island for a livable sized 3 bedroom, expect to pay between HK$45-80k+/ month, depending on location, building age and size. You could find a place for less, but will require more of a search.
Have found that this web site - Hong Kong Property and Apartment, for Rent and for Sale - Hong Kong Realty - gives a pretty good idea of what you can get in different areas. Prices aren't too far off from what you should actually expect to pay and the pictures are okay.
2. Almost everyone uses a Filipina nanny or "helper". We have had generally very good experiences with helpers in Hong Kong. They are often hard working (they will not only care for your children, but also keep the house clean and often can cook) and their English is much better than the average HK locals. Takes some work to find the right lady (she will be living with you, which is required by law, and there is usually a small bedroom and bathroom at the back of the kitchen for her in your apartment) and some getting used to to have someone in your house 24/7, but a good helper is good at knowing when to be around and when not to be. If you give her a TV/ laptop in her room it makes it easy for her to spend time in there relaxing if you want time with just your family.
3. A car not a necessity, unless you live in some places in the New Territories, but even on the South Side of HK island I would probably have one with twin children. Most of my friends from the States are uncomfortable having their kids in taxis without child car seats and child car seats are very difficult to lug around HK, so they tend to have cars used particularly for toting kids about. But, almost all of HK gets by without a car, so it really isn't needed and is a personal preference. The closer you get to Central, the less you would probably want one because more places become walkable, less places become drivable, and there is not a lot of parking. A car is definitely one of those things that you can decide to get after arriving if you feel like your family would benefit from one. Can pick up a safe sedan-type car second hand pretty cheap.