do Centre of gravity and Centre of mass are same?
Generally, Centre of mass and Centre of gravity might not vary a lot. In most of the cases, they coincide about the same point. In most of the engineering problems, the centre of gravity is highly considered over the centre of mass.
why should a body be in equilibrium?
A body which is not in equilibrium will create moments about its center of gravity and therefore create disturbance in itself like rotating, revolving or translates which is not a good thing.
what makes an aircraft leaves typical lines in the sky while traveling?
those are the not the lines but the wing tip vortices that are generated from the wingtips of an aircraft. generally, due to the high pressure of air below the wing and low pressure of the air above the wing, vortices are generated. the mixing of high-pressure air and the low-pressure air turns the air flow and leaves through wingtips as a jet and forms a line in the sky.