A. When the object moves from point C to point D (the direction has not changed), the elastic potential energy of the spring decreases by 3.0J, and the gravitational potential energy of the object increases by 1.0J, so the kinetic energy increases, so the moving direction points to the equilibrium position, so A is correct.

Point B and point D are closer to the equilibrium position, and the resultant force is less than point C, so the acceleration of the object at point C must be greater than that at point D, so B is wrong.

C, in the process of moving from point C to point D (the direction has not changed), the gravitational potential energy of the object has increased by 1.0J, so the position of point C must be below the equilibrium position, so C is wrong.

D, the same kinetic energy corresponds to two positions in the vibration process, which is symmetrical with respect to the equilibrium position, so point D may be below or above the equilibrium position, so the acceleration direction of the object at point C and point D may be opposite, so D is correct.

So choose AD.