You're lucky I'm bored. I guess I'll take a crack at this.
1) Yes, if he works in Sunset Park and you want to be in NYC you must be in Brooklyn. You stress that a 30 min commute is an absolute 100% must, in which case Brooklyn is it.
2) A 4 bedroom 2 bath apartment is probably not happening- there really isn't a ton of rental housing stock that offers space like that. You will definitely be better off renting a house. A brownstone is a type of building. Many were originally built as single family homes and then converted to a couple of apartments later on.
A brownstone that is either not converted or only has one apartment plus a main home would actually be ideal for you. They usually have small yards too. The washer and dryer again would be more likely in a home as opposed to an apartment. I'm not sure what you mean by a 8+ school. NYC has a unique public school system as well as tons and tons of private schools, charter schools, etc. You will need to do a lot of googling on that. $4,000 is doable for a house, but it may not be in the most 'hip' area. It sounds like you're looking for a more family oriented area anyway, which would work to your advantage in terms of price and it being quiet, etc. I would definitely suggest a broker- they will be able to suggest neighborhoods that would work best since you do have quite a few demands.
3) What is included in rent varies widely. Many apartments have heat/hot water included in NYC because they are older (pre-war) apartments, and there is no way to separate each apartment in terms of heat/hot water consumption. i.e. the whole building is on the same meter. Thus, in those buildings the landlord pays. Newer buildings, or those that have been redone top to bottom often do NOT include heat/hot water, as the HVAC/heat/etc has been split by apartment. Heat and hot water is just that. Tenant pays electric bill and gas for gas stoves, etc. But not heating gas costs. Landlords pay water in NYC 99% of the time- again this may differ in a private house. Trash removal is a city service. You can look up online when the trash pickup for your block is. Cable/internet is definitely separate.
4) There is a VERY large Jewish population in Brooklyn- you will definitely be able to find your niche.
5) I honestly have no clue.
6) I think you would he happier in a family-oriented neighborhood, as I said earlier. There are plenty of neighborhoods where you would definitely be among other parents at the park, and there are also plenty that would have more nannies. Places like Park Slope for example, much more likely to have nannies running around.
7) Nope. Most do not. You really really do not need two cars. Whole house rentals might have a small driveway depending on the neighborhood. Craigslist has parking spots for rent. Other than that check for neighborhood garages, but be prepared to pay $300+ a month. Off street parking is DEFINITELY preferable. Even in neighborhoods that have decent street parking your cars will be beat to crap on the street.
8) Grocery shopping is entirely dependent on where you live. Most neighborhoods have decent access to smaller supermarkets, and then each borough also has larger chain grocery stores too (i.e. Trader Joes, Whole Foods). A Whole Foods recently opened in Brooklyn with a large parking lot.
9) There are a couple of hospitals in Brooklyn. I have friends who have had kids at both Maimodities Hospital and Methodist Hospital with no huge complaints. I wouldn't plan on being able to choose your own menu at any of them though.
10) JFK
11) Brooklyn is most definitely not equal in terms of crime, but if you dealt with the things you listed below you will be fine. Again, family oriented neighborhoods will have lower violent crime rates. NYPD will come if you are robbed. Response time for stuff like that isn't too bad. If it's a noise complaint or something of that sort forget about it.
12) Sorry, but this is a dumb question. There is no geographic line. You watch the teams play and pick one. Simple as that.
13) Depends where in Brooklyn and how you try to get there. Some parts 5 mins, some parts 40 mins. I wouldn't drive into Manhattan. Parking in Manhattan, depending on the neighborhood, can easily be $15 an hour. And driving in NYC is a nightmare in general. I would definitely take the train. Costs $2.50 and mass transit is huge here.
14) Yes, Brooklyn has many great neighborhoods with huge food scenes. I would suggest you google it. There are Michelin rated restaurants in Brooklyn.
Smorgasburg is killer too.
15) Yes.
16) I babysat occasionally for two kids (7 and 4) this past year, and was paid $15 an hour. So 6 to midnight you're looking at $90, though their mom would likely give me $100 for that.
17) No. Brooklyn, Manhattan, Williamsburg and Queensboro (I believe?) are free. We have EZPass. Sounds similar.
18) I would suggest craigslist. A friend had a space a couple years ago that he paid $700 a month for. Not sure what's out there these days.