5th avenue is a long long road. I am assuming you are either midtown or downtown.
If uptown, best transportation is either trains to Westchester or Brooklyn. There is Astoria too which I like. Westchester towns - MTA train map - Scarsdale to White Plains and Larchmont to Stamford, CT. Advantage of these areas is that you will get a smaller house with a yard for your current budget.
If downtown, best transportation is either NJ Transit towns (South Orange to Summit or Montclair), Jersey City, Hoboken or Brooklyn. I live in Hoboken and like it a lot. It is very walkable and there are lots of European families here. We used to live in South Orange but the commute killed me and DH found it too suburban.
In Jersey City, Hoboken or Brooklyn you are not going to find a house with a yard. You should be able to find a nice 2-3 bed apartment with a yard and possibly a parking spot at a push.
Also, take a look at what subway would be closest for your DH and go back from there. If Brooklyn is too expensive for you take a look at Park Slope, Carroll Gardens and even as far as Bay Ridge.
I would not sell your house in the UK as you don't have a great visa for the US. I assume you are on a H1B or L2 and, with both of those visas, if you lose your job you have 180 days to leave the country.
For schools go on great schools.org. Tells you about 90% of what you need to know. Narrow it down from there.