Actually, I've lived and worked here in Warsaw since 1997.
Considering tuition in English-speaking schools (I would definitely advize against IAS), it won't be a luxury life but it should be liveable
Yes, after paying rent and school fees, the OP will be left with 1,500zl a month for everything, what a great life that family will have!
I would only want an English maybe Montessori type nursery
Allow 2,500zl per month for that.
I can choose public school for my 2nd grader though I don't know how long it would take her to pick up Polish (we aren't relocating for a long period).
At that age, it'll probably take three years, at least, given the language will be spoken at school only, before the kid is close to the level of his/her peers. Polish state schools provide language help only for a year. Choosing a Polish language school will set your child's education back at least one year, more probably two years.
Electricity and water together wouldn't be more than 200zl a month. But water is usually included in rent anyway.
what are the best neighborhood to live in?
Families seem to like Urszynow, or Mokotow.