Yes, cars do handle better when their lighter... Less wheight means less inertia (stress, pressure, whatever you wanna call it) applied on the tires during cornering. Adding downforce makes handling better because it pushes down on the tires without adding any inertia.
As for the speed, thats because of drag. Most sports cars have a 6th gear thats basically useless when stock. The main purpose of the 6th gear (i think, I could be wrong) is to drop off revs at top speed, and to provide plenty of room to go faster when modified without changing the ratios.