A cambered, or "airfoil-shaped" wing cross section will have a significant curve (bulge) on the top surface, usually with the thickest part nearer the leading edge, while the bottom surface will have no or minimum curve. The result of this is that air passing over the top surface of the airfoil has a longer distance to travel than air passing over the bottom surface. This means air on the top surface flows at a higher relative speed. Since Total air pressure = Static (directly onto the airfoil) Pressure plus Dynamic Pressure (speed of the air), and the Dynamic pressure (speed) on the top is higher, that means to balance the total pressure, the static pressure on the top must be lower. The result of all this head-spinning aerodynamics is that the pressure directly on the bottom (at right angles to) surface of the airfoil is higher than that on the top surface, resulting in aerodynamic lift on that wing (airfoil), even at zero degrees angle of attack. If you inverted the airfoil, so the curved surface was on the bottom, there would be negative lift (downward pressure) at zero degrees angle of attack.
On the other hand, symmetrical wings (airfoils) have no aerodynamic camber, but rather have equal distances for the air to travel over both the top and bottom surfaces. This means they will produce exactly zero lift at zero AOA, and require some angle to produce lift.
Finally, to respond to the true-or-false questions about your two statements, it would depend on each airfoil's curvature. The aerodynamic center of any airfoil will be immediately aft of the point of maximum thickness; on a cambered wing, this will be on the top side, usually well forward of the center point. On a symmetric wing, this will probably be near the center point, and equal on both the top and the bottom.
Many of us have demonstrated this when we were children, by putting our arms out the window of a moving car, and pivoting (pronating and supinating) our hands in the wind to produce up and down lift.