I'm sorry but a 3bed in Brooklyn with decent schools on 40-50K is not going to happen. The nice parts of Brooklyn are insanely expensive and you will have a tough time getting a flat with no credit history as there is so much competition. I can understand not wanting to do NJ but perhaps look at Queens instead?
I think it is possible to be an expat on a normal wage but then you do have to be realistic about what you can get. You wouldn't expect to move to London on 35K and get a 3bed in Zone 2, it's the same idea.
Want is right though, 35K in the UK might translate to 75K in the US, in which case a 2bed in Brooklyn might work.
On the plus side, you work in a great field for NY, however you will be competing with a lot of people willing to work for peanuts (those are popular areas for trust fund kids).
Do NOT move if there's no relocation help whatsoever! I understand not everyone can get the full bells and whistles but they should at least help pay the moving costs.
The insurance question is key, you will probably all be covered but for example it could cost $800 a month, that will make the whole scenario less feasible.
For taxes you pay federal, state and local taxes -- federal is less than in the UK but when you add everything up it's probably the same.
Free kindergarten starts at 5, there is no free childcare. Or actually, I think they started offering free pre-K at 4, but they don't have enough spaces for everyone yet.
I don't want to be a downer because I don't think only rich people should be expats -- we've relocated twice on low incomes. The difference is we went to European countries with lots of social benefits and cheaper living costs. NY is a big shock not only because it's so expensive but because you have to pay for so many more things than you're used to.
So I wouldn't rule it out but see what the salary and insurance costs will be, then you can make a more realistic judgment.
If it doesn't work out, come to Berlin, it's like old school New York and you can still afford to live downtown :)