GA_Driver, welcome to the board. Oh um just a tip, you don't have to hit enter at the end of each line. Keep typing and your post will come out fine.
If you're looking to add a turbo, the RSX may not be the best way to go. Its has a returnless fuel system and the ECU is relatively new and has not been fully cracked. The Integra Type-R is very rare and they command a premium on the used market so I wouldn't bother trying to find one. It's essentially the same thing as an Integra GSR (all the Type-R worshippers are going to chime in here) so if you plan on modifying the car you might as well start with a GSR which is far easier to find and a good deal less expensive.
It does not sound like an RX-7 is the car for you. 1995 was the last model year for them and they require a lot of maintenance and other stuff. Plus when you start modifying them you put yourself at risk of a bunch of other potential problems.
The 240SX has gained a lot of popularity recently and has become very hard to find. In stock form it's not very impressive but you can swap the engine for a Japanese SR20DET (2.0 liter turbo inline four) or slap a turbo on the stock KA24DET 2.4 liter. The suspension and brakes need thorough upgrading as well. There is a huge aftermarket for this car.
The 300ZX Twin Turbo is a freaking boat. It weighs about 3500 pounds. If you don't mind the weight and are looking more for something that can be quick in a straight line and won't mind a spirited drive through the twisties it's a good choice but it's just not my style which is why I have a tough time recommending one. Same deal with the Toyota Supra Turbo, except it's a more modern look both inside and out and can make more horsepower.
If you're not looking to drive some cobbled together "tuner" special I would stay away from getting a Civic hatch. It sounds like an RSX or Integra GSR would be the best choice for you, but given the relative newness of the RSX and the difficulty/expense of tuning it properly for a turbo (people have obviously put turbos on since the car has been around for a little while) I would go for a GSR. There's a number of well known turbo kits available for it and both the fuel system and ECU are known quantities so there isn't much fuss when it comes to getting the car running very nicely. Plus the absolute most you would have to pay for a GSR at this point is maybe $15k or so and that'd be for a really really really really nice 2001 model with low mileage.