1. the sticky part is the thermal pad. unless you plan on overclocking or you think you will be removing and replacing the heasting often, leave the pad on (but make sure you remove the plastic film that usually comes on the pad). you don't use thermal compound at the same time. and yes, this pad makes contact with the little raised square in the middle (that is the core)
2. yes, leave the pads on. they keep the heatsink from tipping or crushing the core.