Can I fit my normal brake levers to the curves of the dropbars?
Sorry, but fitting drop bars to a flat bike is a no-go.

The flat bar levers will not fit properly on a drop bar. You will probably not be able to reach the levers easily which is dangerous ! Reaching the shifters will also be difficult if not impossible.

The other reason is that frames designed for flat bars are longer than those designed for drop bars, as the drop bars extend the hands further forward. You likely find that the hand position on drop bars on a flat bar frame is too far forward.

If you want to get a somewhat faster more aggressive position but retaining you brake lever and shifter hardware, consider replacing your riser bar with a flat, slightly narrower one.