To still fit in the wheel well of the Corolla the tires on a 16inch rim will have shorter sidewalls... i don't think your suspension will suffer, but your ride comfort will be less when going over bumps...
I just had new tires fitted to a Corolla... they were a little expensive at 6,000 per tyre, but they were Yokohama branded, and the improvement in handling was definitely worth it. At least double the grip when braking and high speed cornering. Straight line speed is better too.. with the middle slick strip, it seems to wander across the road much less. They're also 10mm wider than Indus Motors stock tires choice.