1. Get better tires. Compound > size. Not having traction in 3rd gear, at your power level, tells me the grip is lacking mechanically. No suspension will help you with this, unless you compromise your handling to build a drag car.
2. body roll isn't a bad thing. Notice even F1 cars have significant body roll, and the camber to match. The STI felt more confident cornering period, or just only on power while cornering? Chances are your stock dampers are worn out. They wear out very quickly under hard driving, even with totally inexperienced drivers.
3. Coilovers would help address both the look and stability of the car.
Based on what you told me, try this:
CSG Spec Flex A. Stock sways. Better tires.
Lower than your expected budget, addresses everything you listed.
Exactly what I list above was good enough for this yesterday. I run a stock radiator with my turbo, but have trackspec hood louvers. Make your car more efficient, rather than forcing things.
https://www.instagram.com/p/BjlR0C6gCs1