I'm a local resident (Westchester County) and former realtor + I have lived here my entire life so very knowledgeable to answer your question.
1. The issue is renting in a condo can be difficult but not impossible. HOA's can limit or restrict rentals in their buildings to appease lenders. Understand that lenders refuse to finance buyers if the building has a high percentage of renters living there. Every lender is different but they all have a maximum percentage that they will allow. Sellers don't want a feud with you because you renting will prevent them from selling their condo. In the NYC area, condo owners prefer selling vs. renting.
2. Since the job is not confirmed yet, do you know how much advance notice you will have to move? Do you own or rent your condo in Chicago? If you own it, do you need to sell it or do you plan to rent it out?
3. The NYC area is filled with regular rental buildings - even in the suburbs. You will be better off renting in a rental building than a condo building. You need time to explore the different neighborhoods, school districts, etc. Given your price point of $3,500. - 4,000. you can easily find rentals in very nice buildings. The suburbs offer more living space for less money than inside the NYC border.
4. Public Schools - the New York City area is very similar to Chicago. There are low income neighborhoods with poor performing schools like South Chicago and there are very wealthy neighborhoods with top rated school districts like Evanston, Winnetka, etc. The top rated school districts in the New York City suburbs are: Northern New Jersey (from the George Washington Bridge going north up to the New York State Line); in Westchester County - Eastchester, Rye, Irvington, Pelham, Ardsley and Harrison. In Nassau County - Great Neck, Manhasset, Port Washington, the 5 towns area (Hewlett, Woodmere, etc.), Garden City, etc.
New York City - the good neighborhoods to live in are: Bayside, Astoria, Park Slope, Bay Ridge and Staten Island. However, the public school system is very different. Elementary school students attend their neighborhood schools. Middle and High School students must apply to different NYC schools and wait to be accepted. Students don't necessarily attend their neighborhood high school.
5. Like Chicago, the New York City area heavily relies on the subway system and commuter railroads from the suburbs. Public transportation is extensive and excellent. You don't need a car in the city but in the suburbs it is a necessity. There is plenty of public transportation in the suburbs and it's convenient but a car is still worth it.
6. Do you have pets? You will find many rentals and condo buildings are not pet friendly.
7. Beware - Housing is extremely expensive. PLEASE if you accept this position, insist on an income of at least $75,000. Living in the NYC area for less than $75K is going to be a struggle. A financially comfortable income that will meet basic needs is about $125,000.
8. Beware - state income taxes, sales taxes and property taxes (in the suburbs) is extremely high. Therefore, the higher your income, the better off you will be.