Why does the helicopter's nose go down when the cyclic is pressed forward (in a hover)?
I think you might be confused by what the hinges do.  In the example you show, the teetering hinge, which is only present in 2 blade rotors, causes both blades to tilt, or "teeter" or see-saw, together.  It is not possible for both blades to rise or fall in normal operations.  The feathering hinge is not really a hinge at all.  It is a bearing in the blade cuff which enables to blade pitch angle to change, caused by the position of the swash plate, by rotating the blade within it's cuff.

Consider a helicopter hovering in a no wind situation - simply because adding wind needlessly complicates the explanation.  I will also assume a counter-clockwise rotating rotor.  Simply reverse the explanation for a clockwise rotor.

The total thrust vector of the rotor is vertical, perpendicular to the centre axis of the disc.  It has a zero horizontal component so no force is applied longitudinally or laterally to the helicopter and it remains in a fixed position. 

When you push the cyclic forward, the pitch of the blade at the leftmost position is increased, increasing the angle of attack and therefore the lift.  Due to precession, the blade rises at the back of the disc and falls at the front, tilting the entire disc forwards. 

The total thrust vector remains perpendicular to the disc and since the disc is tilted with respect to the fuselage, is now comprised of horizontal and vertical components.  It is the horizontal component which accelerates the helicopter in the direction of the tilt.

So there is a force (lift) lifting the rear of the disc and lowering the front.  This force is transmitted through the rigid rotor drive shaft and pushes the nose down and pulls the tail up. Another way to think about this is that you are applying a force at the top of a lever, the drive shaft, which is above the centre of gravity and therefore causes the fuselage to rotate in pitch. The helicopter will continue to accelerate until the additional drag equals the horizontal component.

Helicopters also employ horizontal stabilisers which produce lift to push the tail down with increasing airspeed so the nose down pitch in cruise is reduced.  

Therefore, pushing the stick forward and holding it at some attitude (when flying VFR you select a speed by adopting the nose down attitude for that speed which is learned from experience and practice followed by small trimming to hit the desired speed) will cause the nose to pitch down and the aircraft to accelerate until the trimmed speed is attained.