Come over here to New York City. It is possibly the most renowned city in the U.S. with many landmarks and things to do and yes, it does get hot during the summer here.
Yep, I was thinking the same thing. Pity Phoenix isn't much of a tourist town, though.
Savannah
Georgia of course!
You could go just about anywhere in the United States during the summer months and get temperatures above 35C (aside Alaska, of course).
If you're serious about the warmth requirement, I'd vote for New Orleans. While it won't quite be <35C (~95F), it can easily be 90F with 90% humidity for most of the summer. There's plenty to do, and now that the streetcars are running along (most of) St. Charles again, it's easy to get from uptown to downtown. The buses also run pretty regularly, and the city is plenty easy to bike around in, there are plenty of options for rentals or picking up a cheap used bike - although in July & August, be prepared to sweat buckets while biking. Downtown is the French Quarter, with Bourbon Street and all that, nightlife aplenty, there's also Frenchman and the Maringny, great for music.
How much you'll need for the month will definitely depend on where you're planning to stay and how extravagantly you're planning to eat - there are plenty of great, cheap places, but there are also plenty of restaurants where you can (easily) spend $50/person, before drinks, for dinner.
I'm at school at Tulane right now, which is in uptown NO, which has it's own set of bars/restaurants/nightlife. There are museums in NO, and plenty of guided tours, etc of the Garden District & Quarter.
Otherwise, if you're just looking to come to the states, I'm from NY and there's plenty to do in NYC for a month - and the options for cheap accommodations are probably a little greater (unless you choose to sublet in NO for a month) as long as you don't mind hostel-type rooms. Public transportation in NYC is the best & cheapest in the world ($2 will take you from the top of the Bronx to Coney island), and there are plenty of museums/monuments/etc in the city to occupy you for a month. Same thing goes for nightlife.
If you want more info, especially on NO, feel free to PM me and I can point you in the direction of more information.
What do you consider "South". I'm from DC, and people here consider themselves in the middle. People from the South think that we're Northerners, and people from the North think that we're Southerners...
DC is a pretty cool town... can be pretty expensive... $2k for a month seems alright, depending on your living situation. That will eat up a lot of your cash.
How old are you, and what do you enjoy doing?