The Tiburon is not in the same class as those other cars. It's more like a competitor for the Civic SI or Mazdaspeed 3 or VW GTI.
The RX-8 is also more finicky about maintenance than the other cars (which you should be properly maintaining anyway). That's a car you only get if you think its unique characteristics are worth the responsibility.
If you can't afford a 350Z, you can't afford a G35 coupe.
The Tiburon, particularly late-model, would be an okay choice. You might also consider its relative, the Elantra. (Not as fast as the Tiburon V6, but it's a pretty good-handling car)
Also consider:
Honda Civic SI/RSX/Integra
Honda Prelude/Accord V6
Toyota Celica
Toyota MR-2
Mazda Miata
Mazda Protege 5/Protege MP3
Subaru Impreza 2.5RS (the WRX might kill you on insurance)
Subaru Forester XT (WRX performance with cheaper insurance)
If you're interested in a bit more torque, and you'd be okay with fuel economy no better (or worse) than the 350Z, you might look into a mid-90s Ford Thunderbird, Mercury Cougar (or if you want more expensive maintenance, and to use premium gas) Lincoln Mark VIII. They're heavy (3700 lbs), but they have V8 torque and fairly responsive (for their size) but very forgiving handling. And the price and insurance is cheap. Or a Mustang or Camaro/Firebird, of course. (Getting a V6 Camaro is okay, getting a V6 Mustang is not, unless it's 05 or later, BTW...but you should only get V8)
The best price/performance/efficiency will be a used Miata or MR-2. (The Miata is far more practical, though...you might think "in that kind of car, who cares about cargo room?" but TRUST ME YOU WILL, especially if it's your only car.)