New York City

I've been offered a transfer from London to NYC, and have been offered a salary of $220k. My wife is going to take a year out of work to care for our infant daughter, so this money will be our full income as a family.

The American system confuses me hugely. Following most calculators, it sounds like we will have roughly $11k a month. If we assume $4k a month on rent somewhere roughly 45-60 mins from Broadway, is $6-7k enough for a family to live on, including stuff like healthcare for the entire family through work?

This might be a silly question, but will we have a comfortable/good time in NYC or NJ on this salary? Will my wife be able to afford to take my daughter out for stuff like swimming lessons and baby classes, and will we have enough to afford stuff like flights home to see family in the UK? I've spoke to a few people at work, and we've had ranges from "wtf half a mil is lots!" to "wtf you'll be limited to NJ and won't be able to take vacations", so it would be good to get more perspectives.