Why is the journey time from London to Singapore less compared to Singapore to London? Is it because of Earth's rotation or because of wind. Why can't one just hover over earth and see earth actually rotating?
It's wind, see this answer: Airplane Travel Times. You take off with whatever speed the Earth's rotation has given you at that particular latitude. But the Earth's surface is also travelling at the same speed. So its only the speed that the plane achieves with its engines, and how local head winds or tail winds affect it, that is important. That's one reason, apart from the obvious ones, why rockets are not launched from the poles, they are launched from the equatorial regions. They use less propellent because they take advantage of the Earth's rotation, which is maximised at the equator.
You can't hover in the way you describe, because the air you are in is moving at the same speed as the Earth, apart from local winds. The Earth turns at 1670 kilometers/hour at the equator. At a latitude of 45 degrees, cos(45) = .707 and the speed is .707 x 1670 = 1180 kilometers/hr. If the air mass was stationary relative to that speed, we would notice it pretty quickly. Don't forget the effect of atmospheric air pressure in a vertical direction.
As per my understanding, the aeroplane will also be dragged by Earth's gravity and so the effect should be negligible. But, is there any difference (however minute) at all caused due to relative speeds and gravitational pull?
There is no gravitional pull of the Earth on the plane, except in a vertical direction. Horizontally, no gravitational force is felt, the aircraft is not "pulled" along, in level flight, by the force of gravity to any measurable extent.
Does the aeroplane require more energy (and fuel) when travelling against the earth's rotation?
No, do you need more energy if you were walking from New York To Los Angeles, or walking the opposite way? In either direction, you use the same energy. Again, it's wind speed that counts far beyond anything else in determining travel time.
When one opens window in a moving car, they can usually feel stronger winds even if the weather outside the car is actually quiet. By same logic, if earth and air are moving in one direction and the aeroplane in another, the wind resistance may certainly be playing some role. Is this a correct assumption? If so, then isn't this apparent wind an indirect effect caused by earth's gravity that drags air along with it(*assuming, weather was otherwise quiet)?
As above, locally wind speeds obviously affect your flight time, but this question is better answered here: Atmosphere And Earth Rotation.. You are forgetting air pressure, which forces the atmosphere onto the surface of the Earth so allowing them both to revolve at the same speed.