I'll tackle this first: No, don't go to a 15" diameter for this car. The sizes that most commonly fit are the 235/60R15 or 245/60R15 (although the 245s, depending on the width of the wheel, could rub.
The best fitment, in my opinion for your car in a 16" diameter would be the 245/50R16. Now, there's not a huge variety, but you can get some well balanced tires for your car. Two tires right off that I would likely recommend if I were to sell them to you, would likely be the BFGoodrich G-Force Super Sport AS or the Michelin Pilot Sport AS Plus. Performance between both tires would compliment what you have already done to the car, without being so aggressive that you break something. And even if you don't plan on driving in wet weather, the siping will allow better cooling efficiency.
What makes the Michelin better than the BFGoodrich, though, is the banded Aramid (also known as Kevlar) strips that are laid over the steel belts. Although more expensive, they don't react to the cold (or hold memory-producing flat spots) like a nylon overlay.
Funny, I haven't seen the Kumho 712 in a long time. I didn't think anyone would still have those on their cars!