A few answers:
If I am be going at all, it will probably be not earlier than November. What I forgot to ask is whether it's at all possible to arrange a trip in such a short notice.
Probably, but try not to schedule your trip around Thanksgiving or Christmas. Travel is insane then.
Re food prices: I don't intend to cook. Cereals for breakfast and sometimes dinner, but I'm almost always going to get lunch outside.
Well if you're willing to buy packaged food like cereals and pop tarts for breakfast then that'll cut down on things. What kind of meals do you want to eat for breakfast and dinner, and how much money are you expecting to spend? Do you insist on drinking beer or wine or coke with your meals, or do you have no problem making do with water?
There are plenty of chain sit down restaurants around here, not fast food but chain restaurants like Applebees or TGI Fridays or Chile's where you can buy decent meals for relatively cheap. (Large $6 burgers with fries, for example.) If you don't mind fast food there there is plenty of variety, and you don't have to eat a McDonalds all the time - there's also Burger King.
(And Wendy's, and KFC, and Arby's, and Subway, and so on)
How much you could reasonably expect to spend on food really depends on what you expect to eat. Steak and lobster isn't going to happen on $20 a day, but if you don't mind eating cereal for breakfast it's definitely possible to eat out on that much the other two meals, and eat decent food, assuming you don't typically drink three or four beers with dinner, and have some cheesecake, and maybe some chicken wings for appetizers....
If you're going to be in a big city, it would probably be more cost effective to just buy a metro ticket than renting a car the whole time. (And then take a train, bus, or rent a car to go to the next city) In DC I think it's about $10-$12 for a day pass, though that's unlimited. There's plenty to see in DC without going on the metro though, if you don't mind walking a lot.
I was in DC with some friends just a few weeks ago, and you can see the Capitol, the Washington Monument, the art museum, the National Archives, and so on without having to drive if you don't want to - you'll just spend a little while hoofing it. Tons of stuff to see. If you don't mind walking though, then you could potentially save a lot of money in DC by doing that. (The viability of that plan would depend on where you're staying, though, and I can't vouch for New York as I've never been there)
I'm afraid I can't help you much on the hotel front if you aren't willing to spend at least $50 a night - try the hostels everyone is linking you to, but be careful, they might not all be....reputable.
If you decide to come visit the US, I hope you have a fun time!