Jembia, here is a great simplification of your helicopter query. Apologies to all the serious helicopter engineers out there, I can't make equations or draw graphs and free body diagrams here
The rotor needs to move as much air, vertically, in the hover or in forward flight, as it takes to support its weight. In the hover, the only source of moving air is the turning blades, which takes engine power. Air with no vertical velocity gets sucked in the top of the disc, and blown out the bottom with some velocity.
Once even a little forward speed is reached, the rotor (which had to be tipped slightly forward to produce the forward speed) also entraps air due to the forward velocity of the tilted disc. The mass flow of air out of the bottom of the disc is now some air moved by the blades (and therefore the engines) and some extra air trapped from the forward motion. This extra air arrives with some velocity, and thus takes less power to be accelerated to the required speed when it comes out of the bottom of the disc. At a very low speed, usually about 3 or 4 knots, this extra air starts to take some portion of the weight of the helicopter, and you need to throttle back if you don't want to climb.
At this very low speed there is very little airframe drag to overcome, so the power required to maintain altitude continues to drop as the forward speed increases. Somewhere around 60 to 80 knots the drag does become significant, and now extra power must be added back to the rotor (from the engines) to keep the total mass of air deflected downwards big enough and fast enough to keep the helicopter from falling AND overcome drag. This continues until you run out of power to add, and you are then at the maximum forward speed of the helicopter - without diving.
WARNING - what is going on with the tail rotor, and the power required to spin a rotor moving at at an angle to the velocity, complicates this simple explanation a bit. But only a bit. ;D
Added in edit: looking at the flying umbrella, the downward moving mass of air is a thin ring coming off the perimeter of the umbrella, while in a helicopter the moving mass of air is a cylinder as big in diameter as the rotor disc. The blades take up a small portion of this disc area, while the umbrella surface area is huge - and heavy. This may explain why the skies are not full of these things today.
Maybe somebody out there with experience in ducted fans can tell us if the same thing happens with a fan buried inside structure?