Wouldn't you be better off flying to Chicago, driving to New York and then flying home from there?
You don't sound like you're planning to make a lot of stops along the way, so you'll probably end up driving there and back on very similar roads - if I were you, I'd save my time and energy for some fun on the backroads, rather than spend all that time on the Interstate.
As for rental companies, check if any of your credit cards/memberhips/other qualifies you for some sort of discount and/or free insurance. I've become a happily loyal Hertz customer because I get complete insurance and (often) a discount through my Hostelling International membership, so if you can find something similar, it can help you.
Cars... I'd recommend following the Way of the May
may
and Jay and getting something that you can actually take to its limits legally and which doesn't scream "STOP ME!!!" at every bored cop. Ditch it as soon as you're done driving across the Verrazano-Narrows Bridge and around Long Island, driving in NYC (particularly Manhattan) is annoying. Oh, and give the Holland Tunnel a try if you want a bit of "this isn't funny anymore".
Attractions: See if the
New York Pass is any good for you; if you want to do and see the things it includes, it can save you a lot of money and also some time. Adding to what's been said and/or is included in the pass, take the Staten Island ferry from Battery Park - to the island and right back again. It's free, relaxing and it gives you a nice view of Lady Liberty.
If you need or want a quiet day out, go to
Broad Channel and marvel at the sleepy little village that it is - right next to JFK and with the skyscrapers of Manhattan nearby. Continue to
Rockaway Park and stick your feet in the Atlantic if you haven't already done so at e.g.
Smith Point.